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Question 1210406: 8^x=16^x+4
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060)   (Show Source):
You can put this solution on YOUR website!

If your problem were 8^x=16^x-4 instead of 8^x=16^x+4 It could be solved approximately on a TI-84 x = 0.85106383 Edwin

Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
8^x = 16^x + 4
~~~~~~~~~~~~~~~~~~~~~

I read this equation exactly and literally as it is written = + . (1) My statement is that this equation has no solutions in real numbers. Indeed, if x >= 0, then, obviously, is less than ; Hence, a fortiori, < + , so equation (1) has no solutions in the domain x >= 0. Next, if x < 0, then left side of equation (1) is less than 1, while right side of this equation is greater than 4, so equation (1) has no solutions in the domain x < 0. Thus the statement is proved and the equation has no solutions in real domain.

Solved.

If, in opposite, your original equation is = , (2) then the reasoning is different and the answer is different, too. Then this equation (2) can be rewritten equivalently in base '2' = , which implies an equation for indexes 3x = 4*(x+4), 3x = 4x + 16, 3x - 4x = 16, -x = 16, x = -16, so the solution for equation (2) is x = -16.

Solved two times for two different interpretations.

By the way, both interpretations lead to meaningful and instructive solutions.

Question 1210407: 8^x=16^x+4
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
8^x = 16^x + 4
~~~~~~~~~~~~~~~~~~~~~

I read this equation exactly and literally as it is written = + . (1) My statement is that this equation has no solutions in real numbers. Indeed, if x >= 0, then, obviously, is less than ; Hence, a fortiori, < + , so equation (1) has no solutions in the domain x >= 0. Next, if x < 0, then left side of equation (1) is less than 1, while right side of this equation is greater than 4, so equation (1) has no solutions in the domain x < 0. Thus the statement is proved and the equation has no solutions in real domain.

Solved.

If, in opposite, your original equation is = , (2) then the reasoning is different and the answer is different, too. Then this equation (2) can be rewritten equivalently in base '2' = , which implies an equation for indexes 3x = 4*(x+4), 3x = 4x + 16, 3x - 4x = 16, -x = 16, x = -16, so the solution for equation (2) is x = -16.

Solved two times for two different interpretations.

Question 1196378: Jonathan obtained his degree and started his own consulting company in January of 2010. At the end of each month he began saving R2 480.00 in his account and requested the bank to quarterly debit three month sum of his exact savings towards his individual retirement account for a total period of ten years. The retirement account earned interest at 11% per annum, compounded quarterly. The amount that was available to him after ten years in the retirement account is
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
Jonathan obtained his degree and started his own consulting company in January of 2010.
At the end of each month he began saving R2 480.00 in his account and requested the bank
to quarterly debit three month sum of his exact savings towards his individual retirement account
for a total period of ten years. The retirement account earned interest at 11% per annum, compounded quarterly.
The amount that was available to him after ten years in the retirement account is
~~~~~~~~~~~~~~~~~~~~~~~~~

The answer R514,664.79 in the post by @ElectricPavlov is INCORRECT.

The correct answer is R530,235.

The formulas in the post by @ElectricPavlov are correct, but his output final answer is wrong.

It is because they systematically use inadequate calculator or inadequate calculation procedure
for their calculations, which does not provide the necessary precision.

So, if you are looking to get a precise answer for your problems in Finance,
which would be correct at the reference level of accuracy for your answer book,
especially for annuity problems, then @ElectricPavlov is a bad source for such a goal.

As I look at their performance, I clearly see that they do not understand, at all,
what approximate calculations are and which requirements such calculations must satisfy.

So, their level of understanding approximate calculations corresponds to a 5th grade school student,
who makes such calculations for the first time in his life and who never took/got lessons
from an experienced mentor on the subject.

But they stick out their chest like a wheel (or as a monument on a pedestal)
and call themselves "artificial intelligence".

A correct name for such a performance is "charlatanism".

Question 1200420: 29% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is (a) exactly three, (b) at least four, (c) less than eight.
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
29% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected.
Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is
(a) exactly three, (b) at least four, (c) less than eight.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the post by @ElectricPavlov, all calculations, all numbers, and all answers are INCORRECT.

The correct answers are:

        (a)   0.246.     (b)   0.4765.     (c)   0.9924.

I checked using the online Binomial distribution calculator

https://stattrek.com/online-calculator/binomial

which is proven to be 100% correct.

Question 1167555: A company that manufactures push scooters has a fixed cost of $10,000 costs $42 produce each scooterThe total cost for the company is the sum of cost and variable castcost function of the number of scooters produced xThen, find C(60) X
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.

C(60) = 10000 + 60*42 dollars.

Use your calculator.

Question 1165912: A company makes pens. They sell each pen for $
5
.
Their revenue is represented by R = 5 x .
The cost to make the pens is $
1
each with a one time start up cost of $
4000
.
Their cost is represented by C = 1 x + 4000 .
a) Find the profit, P, (P = R - C) when the company sells 1000 pens.
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.

They just explained everything to you in the post.   You only need to combine
their explanations in one simple formula, and then calculate

        profit   P = R - C = 5x - (x + 4000) = 5*1000 - (1000+4000) = 5000 - 5000 = 0.

The profit is 0 (zero) at the given conditions.

So, x = 1000 is the break point in this problem.

Question 1171394: 27 % adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is (a) exactly three, (b) at least four, (c) less than eight.
(a) P(3)=
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
27 % adults favor the use of unmanned drones by police agencies.
Twelve U.S. adults are randomly selected. Find the probability
that the number of U.S. adults who favor the use of unmanned drones by police agencies is
(a) exactly three,
(b) at least four,
(c) less than eight.
(a) P(3)=
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Be aware !

In the post by @CPhill, all calculations are INCORRECT.

The correct answers are

(a) 0.2549

(b) 0.4138

(c) 0.9953

Use online free of charge calculator

https://stattrek.com/online-calculator/binomial.aspx

to check your calculations.

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

                Regarding the post by @CPhill . . .

Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.

The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.

                It has no feeling of shame - it is shameless.

This time, again, it made an error.

Although the @CPhill' solutions are copy-paste Google AI solutions, there is one essential difference.

Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.

All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.

Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.

And the last my comment.

When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.

Without it, their reliability is ZERO and their creadability is ZERO, too.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let $X$ be the number of U.S. adults who favor the use of unmanned drones by police agencies.
We are given that the probability of an adult favoring the use of drones is $p = 0.27$.
We have a sample of $n = 12$ adults.
This is a binomial distribution, so $X \sim B(12, 0.27)$.
The probability mass function is given by:
$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$
(a) We want to find $P(X = 3)$, so $k = 3$:
$$ P(X = 3) = \binom{12}{3} (0.27)^3 (1-0.27)^{12-3} $$
$$ P(X = 3) = \binom{12}{3} (0.27)^3 (0.73)^9 $$
$$ \binom{12}{3} = \frac{12!}{3!9!} = \frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1} = 2 \cdot 11 \cdot 10 = 220 $$
$$ P(X = 3) = 220 (0.27)^3 (0.73)^9 $$
$$ P(X = 3) \approx 220 (0.019683) (0.066373) $$
$$ P(X = 3) \approx 220 (0.0013063) \approx 0.287386 $$
$$ P(X = 3) \approx 0.2874 $$
(b) We want to find $P(X \ge 4)$, which is $1 - P(X < 4) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]$.
$$ P(X = 0) = \binom{12}{0} (0.27)^0 (0.73)^{12} \approx 1(1)(0.033626) \approx 0.0336 $$
$$ P(X = 1) = \binom{12}{1} (0.27)^1 (0.73)^{11} \approx 12(0.27)(0.046063) \approx 0.1492 $$
$$ P(X = 2) = \binom{12}{2} (0.27)^2 (0.73)^{10} \approx 66(0.0729)(0.0631) \approx 0.3039 $$
$$ P(X < 4) = 0.0336 + 0.1492 + 0.3039 + 0.2874 = 0.7741 $$
$$ P(X \ge 4) = 1 - 0.7741 = 0.2259 $$
(c) We want to find $P(X < 8) = 1 - P(X \ge 8) = 1 - [P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)]$.
Alternatively, we can compute $P(X < 8) = \sum_{k=0}^{7} P(X=k)$.
Using a calculator or software:
$$ P(X < 8) \approx 0.9997 $$
Using the complement, $P(X \ge 8)$:
$$ P(X=8) \approx 0.00021 $$
$$ P(X=9) \approx 0.00002 $$
$$ P(X=10) \approx 0.000001 $$
$$ P(X=11) \approx 0.00000003 $$
$$ P(X=12) \approx 0.0000000002 $$
Summing those gives $P(X\ge 8) \approx 0.00023$.
$P(X<8) = 1 - 0.00023 = 0.99977 \approx 0.9998$.
Final Answer: The final answer is $\boxed{0.2874, 0.2259, 0.9998}$

Question 1167931: Use the upper and lower bound theorem find the smallest positive integer and largest negative integer that are upper and lower bounds , respectively for the real zeros of (B ) Approximate the real zeros of each polynomial to two P(x) decimal places .
P(x) = x ^ 4 - x ^ 3 - 8x ^ 2 - 12x - 25
Found 3 solutions by Edwin McCravy, ikleyn, mccravyedwin:
Answer by Edwin McCravy(20060)   (Show Source):
You can put this solution on YOUR website!

>In his post, Edwin AGAIN makes lunge toward me, without any visible reason/cause from my side.<< I made no lunge at you, Ikleyn, where did you get that? I merely said that you complain about the AI "tutor", which you do because it has not learned everything in mathematics yet. Then I said "but in 10 years it'll be able to solve anything humans can solve." And it will be, and will take over many jobs. You recently referred to something AI had written as "gibberish". Is that not complaining? >>>Edwin, what I do for this AI by checking their errors, and what I do for visitors of this forum, explaining/revealing the errors of the AI, is INVALUABLE SERVICE for them. But you either do not understand it (unfortunately), or deliberately try to throw black paint on what I do at this forum. Edwin, let me tell you how an intelligent gentleman would review my work: "Thanks to tutor @ikleyn for her tireless daily work on finding, identifying and fixing the errors made by artificial intelligence. This work brings closer the day when artificial intelligence will become a reliable assistant to teachers and students in their daily work." It is a true intelligent form and a true intelligent style of writing a review, and it is a style of how a true professor should write. I decided to share this " HOW TO " with you, in case if you are unfamiliar with it. Also, @ikleyn does not complain about Artificial Intelligence (because ikleyn is not an idiot to complain about AI). Are you denying recently calling the AI's output "gibberish"? @ikleyn makes hidden errors, made by the AI, visible and fixes them, if to tell truth. Edwin, English is your native language, isn't it ? And you are a professor, right ? Yes, I was a mathematics prof for 40 years, from 1966 through 2005. I am now 88 years of age. Then why it is so difficult for you to tell truth in its proper and appropriate form, using right terms ? What falsehood do you think I told? I said that in 10 years AI will be able to solve anything. I stand by that. Edwin

Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.

        In his post, Edwin AGAIN makes lunge toward me, without any visible reason/cause from my side.

        Edwin, what I do for this AI by checking their errors,
        and what I do for visitors of this forum, explaining/revealing
        the errors of the AI, is INVALUABLE SERVICE for them.

        But you either do not understand it (unfortunately),
        or deliberately try to throw black paint on what I do at this forum.

Edwin, let me tell you how an intelligent gentleman would review my work:

    "Thanks to tutor @ikleyn for her tireless daily work on finding, identifying and fixing
    the errors made by artificial intelligence.
    This work brings closer the day when artificial intelligence will become a reliable assistant
    to teachers and students in their daily work."

It is a true intelligent form and a true intelligent style of writing a review,
and it is a style of how a true professor should write.
I decided to share this " HOW TO " with you, in case if you are unfamiliar with it.

Also, @ikleyn does not complain about Artificial Intelligence
(because ikleyn is not an idiot to complain about AI).

@ikleyn makes hidden errors, made by the AI, visible and fixes them, if to tell truth.

Edwin, English is your native language, isn't it ?   And you are a professor, right ?

Then why it is so difficult for you to tell truth in its proper and appropriate form, using right terms ?

Answer by mccravyedwin(408)   (Show Source):
You can put this solution on YOUR website!

Here's a problem somebody posted 5 years ago in 2020 1167931 (2020-10-22 08:58:07): Use the upper and lower bound theorem find the smallest positive integer and largest negative integer that are upper and lower bounds , respectively for the real zeros of Notice the rows of numbers at the bottom of the long divisions by -3 through 5: -3|1 -1 -8 -12 -25 | -3 12 -12 72 1 -4 4 -24 47 <-- notice that the signs on the bottom row of the synthetic division DO INDEED alternate. That means that -3 IS INDEED a lower bound for negative zeros. -2|1 -1 -8 -12 -25 | -2 6 4 16 1 -3 -2 -8 -9 <-- notice that the signs on the bottom row of the synthetic division DO NOT alternate. That means that -2 IS NOT a lower bound for negative zeros. -1|1 -1 -8 -12 -25 | -1 2 6 6 1 -2 -6 -6 -19 <-- notice that the signs on the bottom row of the synthetic division DO NOT alternate. That means that -1 IS NOT a lower bound for negative zeros. 0|1 -1 -8 -12 -25 | 0 0 0 0 1 -1 -8 -12 -25 <-- notice that the signs on the bottom row of the synthetic division DO NOT alternate nor are they all the same. That means that 0 IS NOT a lower bound for negative zeros nor an upper bound for positive zeros. 1|1 -1 -8 -12 -25 | 1 0 -8 -20 1 0 -8 -20 -45 <-- notice that the signs on the bottom row of the synthetic division are NOT all the same. That means that 1 IS NOT an upper bound for positive zeros. 2|1 -1 -8 -12 -25 | 2 2 -12 -48 1 1 -6 -24 -73 <-- notice that the signs on the bottom row of the synthetic division are NOT all the same. That means that 2 IS NOT an upper bound for positive zeros. 3|1 -1 -8 -12 -25 | 3 6 -6 -54 1 2 -2 -18 -79 <-- notice that the signs on the bottom row of the synthetic division are NOT all the same. That means that 3 IS NOT an upper bound for positive zeros. 4|1 -1 -8 -12 -25 | 4 12 16 16 1 3 4 4 -9 <-- notice that the signs on the bottom row of the synthetic division are NOT all the same. That means that 4 IS NOT an upper bound for positive zeros. 5|1 -1 -8 -12 -25 | 5 20 60 240 1 4 12 48 215 <-- notice that the signs on the bottom row of the synthetic division are ALL THE SAME. That means that 5 IS INDEED an upper bound for positive zeros. I don't know if this upper and lower bound theorem for roots or zeros of polynomials is taught anymore. Apparently it still was back in 2020. Anyway the largest negative integer that is a lower bound for negative zeros is -3 and the smallest positive integer that is an upper bound for positive zeros is 5. I'm not going to do part (B) for it's long and involved and done with synthetic division. I just wondered if they bother teaching these long methods anymore, since calculators and computers can find them so quickly. Ikleyn complains about the AI "tutor", but in 10 years it'll be able to solve anything humans can solve. Edwin

Question 1180634: This pandemic, Abheedette learned to bake while on home quarantine. She
also realized that she will be able to make P60.00 profit per tray of banana muffins and P120.00 profit per tray of blueberry muffins. She needs 2 cups of milk and 3 cups of flour to bake a tray of banana muffins. And, baking a tray of blueberry muffins takes 4 cups of milk and 3 cups of flour. She has 16 cups of milk and 15 cups of flour. How many trays of each flavor must be baked to maximize the profit?
a. Define the variable used:
b. LP Model:
c. Identify the feasible region:
d. Corner Points and the objective functions:
e. Optimal Solution (final answer):
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
This pandemic, Abheedette learned to bake while on home quarantine.
She also realized that she will be able to make P60.00 profit per tray of banana muffins
and P120.00 profit per tray of blueberry muffins.
She needs 2 cups of milk and 3 cups of flour to bake a tray of banana muffins.
And, baking a tray of blueberry muffins takes 4 cups of milk and 3 cups of flour.
She has 16 cups of milk and 15 cups of flour.
How many trays of each flavor must be baked to maximize the profit?
a. Define the variable used:
b. LP Model:
c. Identify the feasible region:
d. Corner Points and the objective functions:
e. Optimal Solution (final answer):
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @CPhill is INCORRECT.
        His answer x= 3, y= 3 does not satisfy the constraints.
        So, I came to bring you a correct solution.

**a. Define the variables used:** * Let `x` be the number of trays of banana muffins. * Let `y` be the number of trays of blueberry muffins. **b. LP Model:** * **Objective function:** Maximize profit (P) = 60x + 120y * **Constraints:** * Milk: 2x + 4y ≤ 16 * Flour: 3x + 3y ≤ 15 * Non-negativity: x ≥ 0, y ≥ 0 **c. Identify the feasible region:** To graph the feasible region, we'll first find the intercepts of the constraint lines: * **Milk constraint (2x + 4y ≤ 16):** * x-intercept (y = 0): 2x = 16 => x = 8 * y-intercept (x = 0): 4y = 16 => y = 4 * **Flour constraint (3x + 3y ≤ 15):** * x-intercept (y = 0): 3x = 15 => x = 5 * y-intercept (x = 0): 3y = 15 => y = 5 Now, plot these lines and shade the region that satisfies all constraints (including non-negativity). Find the intersection point (x,y) = (2,3) of the sloping lines. The feasible region is a polygon with vertices at (0, 0), (0, 4), (2, 3), and (5, 0). See my plot under this link https://www.desmos.com/calculator/hxbuw3toqz **d. Corner Points and the objective functions:** Evaluate the objective function (P = 60x + 120y) at each corner point: * (0, 0): P = 60(0) + 120(0) = 0 * (0, 4): P = 60(0) + 120(4) = 480 * (2, 3): P = 60(2) + 120(3) = 480 * (5, 0): P = 60(5) + 120(0) = 300 **e. Optimal Solution (final answer):** There are two optimal points: (0,4) and (2,3). It means that the maximum profit of P480.00 is achieved when Abheedette bakes 0 trays of banana muffins and 4 trays of blueberry muffins, or 2 trays of banana muffins and 3 trays of blueberry muffins. It is a rare case in solving minimax linear problems, when where are more that one solution. It may happen when the slope of one of boundary lines coincides with the gradient of the objective function. It is a rare case, but, nevertheless, it may happen, as you see it in this problem.

Solved.

For plotting, I used free of charge plotting tool at this site http:\\www.desmos.com/calculator/

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's the solution to maximize Abheedette's profit:
**a. Define the variables used:**
* Let `x` be the number of trays of banana muffins.
* Let `y` be the number of trays of blueberry muffins.
**b. LP Model:**
* **Objective function:** Maximize profit (P) = 60x + 120y
* **Constraints:**
* Milk: 2x + 4y ≤ 16
* Flour: 3x + 3y ≤ 15
* Non-negativity: x ≥ 0, y ≥ 0
**c. Identify the feasible region:**
To graph the feasible region, we'll first find the intercepts of the constraint lines:
* **Milk constraint (2x + 4y ≤ 16):**
* x-intercept (y = 0): 2x = 16 => x = 8
* y-intercept (x = 0): 4y = 16 => y = 4
* **Flour constraint (3x + 3y ≤ 15):**
* x-intercept (y = 0): 3x = 15 => x = 5
* y-intercept (x = 0): 3y = 15 => y = 5
Now, plot these lines and shade the region that satisfies all constraints (including non-negativity). The feasible region is a polygon with vertices at (0, 0), (0, 4), (3, 3), and (5, 0).
**d. Corner Points and the objective functions:**
Evaluate the objective function (P = 60x + 120y) at each corner point:
* (0, 0): P = 60(0) + 120(0) = 0
* (0, 4): P = 60(0) + 120(4) = 480
* (3, 3): P = 60(3) + 120(3) = 540
* (5, 0): P = 60(5) + 120(0) = 300
**e. Optimal Solution (final answer):**
The maximum profit of P540.00 is achieved when Abheedette bakes 3 trays of banana muffins and 3 trays of blueberry muffins.

Question 1168845: A survey of licensed drivers inquired about running red lights. One question asked, "Of every 10 motorists who run a red light, about how many do you think will be caught?" The mean result for 880 respondents was
¯x= 1.92. Suppose we know that
σ= 1.83. Compute a 95% confidence interval for the mean opinion in the population of all licensed drivers. (Round your answer to the nearest hundredth.)
(,)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's calculate the 95% confidence interval for the mean opinion.
**Given Information:**
* Sample mean (x̄) = 1.92
* Population standard deviation (σ) = 1.83
* Sample size (n) = 880
* Confidence level = 95%
**Steps:**
1. **Find the Z-score for the Confidence Level:**
* For a 95% confidence interval, the z-score is 1.96. This value corresponds to the z-scores that capture 95% of the area under the standard normal distribution.
2. **Calculate the Standard Error:**
* Standard error (SE) = σ / √n
* SE = 1.83 / √880
* SE ≈ 1.83 / 29.66479
* SE ≈ 0.06168
3. **Calculate the Margin of Error:**
* Margin of error (ME) = z * SE
* ME = 1.96 * 0.06168
* ME ≈ 0.12089
4. **Calculate the Confidence Interval:**
* Confidence interval = x̄ ± ME
* Lower bound = x̄ - ME = 1.92 - 0.12089 ≈ 1.79911
* Upper bound = x̄ + ME = 1.92 + 0.12089 ≈ 2.04089
5. **Round to the Nearest Hundredth:**
* Lower bound ≈ 1.80
* Upper bound ≈ 2.04
**Therefore, the 95% confidence interval for the mean opinion in the population of all licensed drivers is (1.80, 2.04).**

Question 1168930: You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately
σ=69.7.You would like to be 99.5% confident that your estimate is within 2 of the true population mean. How large of a sample size is required?
Do not round mid-calculation. However, use a critical value accurate to three decimal places — this is important for the system to be able to give hints for incorrect answers.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
To determine the required sample size, we can use the formula for the margin of error when the population standard deviation is known:
$E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$
where:
$E$ is the desired margin of error (within 2)
$z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level
$\sigma$ is the population standard deviation (69.7)
$n$ is the required sample size
We are given a 99.5% confidence level. The significance level $\alpha$ is:
$\alpha = 1 - 0.995 = 0.005$
The critical value $z_{\alpha/2}$ corresponds to the $1 - \alpha/2 = 1 - 0.005/2 = 1 - 0.0025 = 0.9975$ percentile of the standard normal distribution.
We need to find the z-score such that the area to the left is 0.9975. Using a standard normal distribution table or a calculator, we find the z-score:
$z_{0.9975} \approx 2.807$
Now, we can plug the values into the margin of error formula and solve for $n$:
$E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$
$2 = 2.807 \times \frac{69.7}{\sqrt{n}}$
Now, solve for $\sqrt{n}$:
$\sqrt{n} = \frac{2.807 \times 69.7}{2}$
$\sqrt{n} = \frac{195.6479}{2}$
$\sqrt{n} = 97.82395$
Square both sides to find $n$:
$n = (97.82395)^2$
$n \approx 9569.52$
Since the sample size must be a whole number, we round up to the nearest integer to ensure the desired margin of error is met.
$n = 9570$
Final Answer: The final answer is $\boxed{9570}$

Question 1170424: 𝐏𝐫𝐨𝐛𝐥𝐞𝐦 𝟏

(a) Teddy J is a manufacturer of dish washing liquid . If his monthly demand function for 750ml size is q = 4000 − 250p and his total cost function is C(q) = 500 + 0.2q.
(i) Derive an expression, R(q) for Teddy J′s total revenue curve.
(ii) Derive an expression, Π(q) for Teddy J′s profit function.
(iii) Determine whether Teddy J′s profit is increasing or decreasing when he produces 5 hundred, 750ml bottles of dish washing liquid.
(iv) How many 750ml bottles of dish washing liquid should Teddy J produce per month if he wishes to maximize his profits.
(b) A firm has an average cost function
125 q2
A(q) = + − 4. q 16
where q is the firm′s output.
(i) Determine the level of output for average costs are minimum.
(ii) Hence determine the range of values for which average costs are decreasing.
(iii) What part of the decreasing range is practically feasible?
(iv) Write an equation for the total cost function.
(v) Hence calculate the level of output for which total costs are minimum.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down each part of the problem.
**(a) Teddy J's Dish Washing Liquid**
Given:
* Demand function: $q = 4000 - 250p$
* Total cost function: $C(q) = 500 + 0.2q$
**(i) Total Revenue (R(q))**
* First, solve the demand function for $p$:
* $250p = 4000 - q$
* $p = \frac{4000 - q}{250} = 16 - \frac{q}{250}$
* Total revenue is given by $R(q) = pq$:
* $R(q) = \left(16 - \frac{q}{250}\right)q$
* $R(q) = 16q - \frac{q^2}{250}$
**(ii) Profit Function (Π(q))**
* Profit is total revenue minus total cost:
* $\Pi(q) = R(q) - C(q)$
* $\Pi(q) = \left(16q - \frac{q^2}{250}\right) - (500 + 0.2q)$
* $\Pi(q) = 16q - \frac{q^2}{250} - 500 - 0.2q$
* $\Pi(q) = -\frac{q^2}{250} + 15.8q - 500$
**(iii) Profit at q = 500**
* To determine if profit is increasing or decreasing, we need to find the derivative of the profit function and evaluate it at $q = 500$:
* $\Pi'(q) = -\frac{2q}{250} + 15.8 = -\frac{q}{125} + 15.8$
* $\Pi'(500) = -\frac{500}{125} + 15.8 = -4 + 15.8 = 11.8$
* Since $\Pi'(500) > 0$, the profit is increasing when Teddy J produces 500 bottles.
**(iv) Profit Maximization**
* To maximize profit, we set the derivative of the profit function to zero:
* $\Pi'(q) = -\frac{q}{125} + 15.8 = 0$
* $\frac{q}{125} = 15.8$
* $q = 15.8 \cdot 125 = 1975$
* Teddy J should produce 1975 bottles to maximize profits.
**(b) Firm's Average Cost Function**
Given:
* Average cost function: $A(q) = \frac{125}{q} + \frac{q}{16} - 4$
**(i) Output for Minimum Average Costs**
* To find the minimum average cost, we take the derivative of $A(q)$ and set it to zero:
* $A'(q) = -\frac{125}{q^2} + \frac{1}{16} = 0$
* $\frac{125}{q^2} = \frac{1}{16}$
* $q^2 = 125 \cdot 16 = 2000$
* $q = \sqrt{2000} = 20\sqrt{5} \approx 44.72$
* The level of output for minimum average costs is approximately 44.72.
**(ii) Range for Decreasing Average Costs**
* Average costs are decreasing when $A'(q) < 0$:
* $-\frac{125}{q^2} + \frac{1}{16} < 0$
* $\frac{1}{16} < \frac{125}{q^2}$
* $q^2 < 125 \cdot 16 = 2000$
* $q < \sqrt{2000} = 20\sqrt{5} \approx 44.72$
* Average costs are decreasing when $0 < q < 20\sqrt{5}$.
**(iii) Practically Feasible Decreasing Range**
* Output must be a positive value.
* The practically feasible range for decreasing average costs is $0 < q < 20\sqrt{5}$.
**(iv) Total Cost Function (C(q))**
* Total cost is average cost multiplied by output:
* $C(q) = q \cdot A(q)$
* $C(q) = q \left(\frac{125}{q} + \frac{q}{16} - 4\right)$
* $C(q) = 125 + \frac{q^2}{16} - 4q$
**(v) Output for Minimum Total Costs**
* To find the minimum total cost, we take the derivative of $C(q)$ and set it to zero:
* $C'(q) = \frac{2q}{16} - 4 = \frac{q}{8} - 4 = 0$
* $\frac{q}{8} = 4$
* $q = 32$
* The level of output for minimum total costs is 32.

Question 1170427: 𝐏𝐫𝐨𝐛𝐥𝐞𝐦 𝟒

(b) During a nationwide program to immunize the population against a new strain of the flu, public health officials determined that the cost of inoculating x% of the susceptible population
1.85x
would be approximately C(x) = million dollars. x
(i) What would it cost to providing immunization to the first 20% of the susceptible population?
(ii) What would it cost to providing immunization to the next 30% of the susceptible population?
(iii) Suppose 17 million dollars are available for providing immunization. What percentage of the susceptible population will not receive immunization?
(iv) If money was not a problem will they be able to providing immunization to the entire susceptible population?

(c) Determine the values of x for which the function f

is discontinuous.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down each part of the problem.
**(b) Immunization Costs**
We are given the cost function:
$$C(x) = \frac{1.85x}{100-x}$$
where $x$ is the percentage of the susceptible population immunized, and $C(x)$ is in million dollars.
**(i) Cost for the first 20%:**
We need to find $C(20)$.
$$C(20) = \frac{1.85(20)}{100-20} = \frac{37}{80} = 0.4625$$
So, it would cost 0.4625 million dollars, or $462,500.
**(ii) Cost for the next 30%:**
If we immunize the first 20%, then we need to immunize from 20% to 50%.
First, find the cost of immunizing 50%:
$$C(50) = \frac{1.85(50)}{100-50} = \frac{92.5}{50} = 1.85$$
The cost for the next 30% is $C(50) - C(20)$:
$$1.85 - 0.4625 = 1.3875$$
So, it would cost 1.3875 million dollars, or $1,387,500.
**(iii) Percentage not immunized with $17 million:**
We need to find $x$ such that $C(x) = 17$:
$$17 = \frac{1.85x}{100-x}$$
$$17(100-x) = 1.85x$$
$$1700 - 17x = 1.85x$$
$$1700 = 18.85x$$
$$x = \frac{1700}{18.85} \approx 90.1856764$$
So, approximately 90.19% of the population can be immunized.
The percentage not immunized is $100 - x$:
$$100 - 90.19 \approx 9.81$$
Approximately 9.81% of the population will not receive immunization.
**(iv) Immunization for the entire population:**
If we want to immunize 100% of the population, we need to find $C(100)$:
$$C(100) = \frac{1.85(100)}{100-100} = \frac{185}{0}$$
This is undefined, as we cannot divide by zero.
Therefore, it is impossible to immunize the entire susceptible population.
**(c) Discontinuity of f(x)**
You didn't provide the function f(x). However, I can explain how to find the discontinuities of a function.
**Finding Discontinuities:**
* **Rational Functions:** If $f(x)$ is a rational function (a fraction where the numerator and denominator are polynomials), then the function is discontinuous where the denominator is zero.
* **Piecewise Functions:** If $f(x)$ is a piecewise function, then the function is discontinuous where the pieces do not meet.
* **Other Functions:** If $f(x)$ involves other functions like logarithms or square roots, then the function is discontinuous where those functions are undefined.
**Example:**
Let's say $f(x) = \frac{x+1}{x-2}$.
To find the discontinuities, we set the denominator to zero:
$x - 2 = 0$
$x = 2$
Therefore, $f(x)$ is discontinuous at $x = 2$.
**Please provide the function f(x) so I can determine its discontinuities.**

Question 1170621: Consider the following system
𝑥1 − 𝑥2 + 3𝑥3 − 𝑥4 = 0
𝑥1 + 4𝑥2 − 𝑥3 + 𝑥4 = 3
3𝑥1 + 7𝑥2 + 𝑥3 + 𝑥4 = 6
Is there any successful decomposition into endogenous and exogeneous variables? Explain.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
To determine if there's a successful decomposition into endogenous and exogenous variables, we need to analyze the system of equations and see if we can express some variables as functions of the others.
The system of equations is:
1. $x_1 - x_2 + 3x_3 - x_4 = 0$
2. $x_1 + 4x_2 - x_3 + x_4 = 3$
3. $3x_1 + 7x_2 + x_3 + x_4 = 6$
We have 3 equations and 4 variables ($x_1, x_2, x_3, x_4$). This means we have an underdetermined system, and there are infinitely many solutions (or no solutions, but we can find some).
**Endogenous and Exogenous Variables:**
* **Endogenous variables:** Variables whose values are determined within the system.
* **Exogenous variables:** Variables whose values are determined outside the system.
A successful decomposition requires that we can express some variables (endogenous) as functions of the remaining variables (exogenous).
**Analysis:**
1. **Solve for $x_1$ from equation 1:**
* $x_1 = x_2 - 3x_3 + x_4$
2. **Substitute this into equations 2 and 3:**
* Equation 2: $(x_2 - 3x_3 + x_4) + 4x_2 - x_3 + x_4 = 3$
* $5x_2 - 4x_3 + 2x_4 = 3$
* Equation 3: $3(x_2 - 3x_3 + x_4) + 7x_2 + x_3 + x_4 = 6$
* $10x_2 - 8x_3 + 4x_4 = 6$
* $5x_2 - 4x_3 + 2x_4 = 3$
3. **Notice that the resulting equations from 2 and 3 are identical.** This means we have effectively only two independent equations.
**Conclusion:**
* We can express $x_1$ as a function of $x_2, x_3, x_4$.
* We can express $5x_2 - 4x_3 + 2x_4 = 3$, which gives us one relationship between $x_2, x_3, x_4$.
Therefore, we can successfully decompose the variables.
* **Endogenous:** $x_1, x_2$ (or any two variables)
* **Exogenous:** $x_3, x_4$ (or the remaining two variables)
**Explanation:**
We can choose any two variables as exogenous. For example, if we let $x_3$ and $x_4$ be exogenous, we can solve for $x_2$ and then $x_1$.
1. $5x_2 = 4x_3 - 2x_4 + 3$
2. $x_2 = \frac{4}{5}x_3 - \frac{2}{5}x_4 + \frac{3}{5}$
3. $x_1 = (\frac{4}{5}x_3 - \frac{2}{5}x_4 + \frac{3}{5}) - 3x_3 + x_4$
4. $x_1 = -\frac{11}{5}x_3 + \frac{3}{5}x_4 + \frac{3}{5}$
Therefore, $x_1$ and $x_2$ are successfully expressed as functions of $x_3$ and $x_4$.
**Final Answer:** Yes, there is a successful decomposition into endogenous and exogenous variables. For example, we can choose $x_1$ and $x_2$ as endogenous and $x_3$ and $x_4$ as exogenous.

Question 1170795: a person pushes a 20 kg shopping car at a constant velocity for a distance of 33m. she pushes the handle in a direction 30 degrees to the horizontal. A 52N frictional force opposes the motion of the cart.
how large is the force in the handle that pushes the cart forward (60.04N)
determine the work done by
a) the pushing force (1.72*10^3J)
b) the frictional force (-1.72*10^3J)
c) the gravational force (0J)
could u show me steps on solving!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step by step.
**1. Finding the Pushing Force**
* **Forces Involved:**
* Applied force (F) at 30 degrees to the horizontal.
* Frictional force (f) = 52 N, opposing motion.
* Since the cart moves at a constant velocity, the net force in the horizontal direction is zero.
* **Horizontal Component of Applied Force:**
* F_x = F * cos(30°)
* **Equilibrium Condition:**
* F_x = f
* F * cos(30°) = 52 N
* **Solving for F:**
* F = 52 N / cos(30°)
* F = 52 N / (√3 / 2)
* F ≈ 60.04 N
**2. Work Done by the Pushing Force**
* **Work Done Formula:**
* W = F * d * cos(θ)
* Where:
* W is work done.
* F is the force.
* d is the distance.
* θ is the angle between the force and the displacement.
* **Calculation:**
* W = 60.04 N * 33 m * cos(30°)
* W ≈ 1716.9 J
* W ≈ 1.72 × 10³ J
**3. Work Done by the Frictional Force**
* **Frictional Force:**
* f = 52 N, opposing motion.
* **Work Done by Friction:**
* W_f = f * d * cos(180°)
* The angle between the frictional force and the displacement is 180 degrees because they are in opposite directions.
* W_f = 52 N * 33 m * (-1)
* W_f = -1716 J
* W_f ≈ -1.72 × 10³ J
**4. Work Done by the Gravitational Force**
* **Gravitational Force:**
* Fg = mg (where m is mass and g is acceleration due to gravity).
* **Work Done by Gravity:**
* W_g = Fg * d * cos(θ)
* The gravitational force acts vertically downwards, while the displacement is horizontal.
* Therefore, the angle between the gravitational force and the displacement is 90 degrees.
* W_g = Fg * d * cos(90°)
* W_g = Fg * d * 0
* W_g = 0 J
**Summary of Answers:**
* **Pushing Force:** 60.04 N
* **Work Done by Pushing Force:** 1.72 × 10³ J
* **Work Done by Frictional Force:** -1.72 × 10³ J
* **Work Done by Gravitational Force:** 0 J

Question 1209931: Find x, giving your answer in the form a + square root(b)
2 = x(squareroot(7)-3)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's solve for x:
2 = x(√7 - 3)
To isolate x, we need to divide both sides by (√7 - 3):
x = 2 / (√7 - 3)
Now, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of (√7 - 3), which is (√7 + 3):
x = 2(√7 + 3) / [(√7 - 3)(√7 + 3)]
Now, let's simplify the denominator:
(√7 - 3)(√7 + 3) = (√7)² - (3)² = 7 - 9 = -2
Now, substitute this back into the expression for x:
x = 2(√7 + 3) / (-2)
Divide both the numerator and the denominator by 2:
x = (√7 + 3) / (-1)
x = -√7 - 3
Rearrange to the desired form a + √b:
x = -3 - √7
Therefore, x = -3 - √7.

Question 1209932: Using the formula v=1/3πr^2H how do I write a simplified algebraic expression for the volume of a cone with a radius of ab^3 and a height of 6b/a?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down the problem and substitute the given values into the formula.
Formula:
v = (1/3)πr²H
Given Values:
Radius (r) = ab³
Height (H) = 6b/a
Substitute the Values:
v = (1/3)π(ab³)²(6b/a)
Simplify the Expression:
Square the radius:
(ab³)² = a²b⁶
Substitute the squared radius back into the volume formula:
v = (1/3)π(a²b⁶)(6b/a)
Multiply the terms:
v = (1/3)π * 6 * (a²b⁶ * b / a)
Simplify the numerical terms:
(1/3) * 6 = 2
v = 2π(a²b⁶ * b / a)
Simplify the variables:
a²/a = a
b⁶ * b = b⁷
v = 2πab⁷
Simplified Algebraic Expression:
The simplified algebraic expression for the volume of the cone is:
v = 2πab⁷

Question 1173170: 5. What sum must be deposited today at 18% per year compounded monthly if the goal is to a compound amount of $50, 00 six years from today? How much interest will be earned during this period?
Question no 03:
1. Determine the present value of a series of 60 monthly payments of $2,500 each which begins 1 month from today. Assume interest of 12 percent per year compounded monthly.
2.
3.
Determine the present value of a series of 36 monthly payments of $5,000 each which
begins 1 month from today. Assume interest of 18 percent per year compounded
monthly
A person wants to buy a life insurance policy which would yield a large enough sum of money to provide for 20 annual payments of $50,000 to surviving members of the family. The payments would begin 1 year from the time of death. It is assumed that interest could be earned on the sum received from the policy at a rate of 8 percent per year compounded annually.
(a) What amount of insurance should be taken out so as to ensure the desired annuity? (6) How much interest will be earned on the policy benefits over the 20-year period?
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.

                Regarding the post by @CPhill:

Do not trust to calculations by @CPhill: I checked them,

and I found that they all are from to be accurate,

from to be precise and from to be correct.

                Regarding your post:

NEVER submit more than one problem/question per post.

It is PROHIBITED by the rules of this forum,

and those who VIOLATE these rules, work AGAINST their own interests.

******************************************************************

    In other words, it is a greatest degree of stupidity
        to post many problems/questions in one post.

******************************************************************

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's tackle each of these financial calculations step-by-step.
**Question 1: Lump Sum Deposit for Future Value**
1. **Identify the variables:**
* Future value (FV): $50,000
* Interest rate per year (r): 18% or 0.18
* Compounding frequency (n): Monthly, so 12 times per year
* Interest rate per period (i): r/n = 0.18 / 12 = 0.015
* Number of years (t): 6
* Total number of periods (N): t * n = 6 * 12 = 72
2. **Use the present value formula:**
* PV = FV / (1 + i)^N
3. **Plug in the values:**
* PV = $50,000 / (1 + 0.015)^72
* PV = $50,000 / (1.015)^72
* PV = $50,000 / 2.930491879
* PV = $17,061.85 (approximately)
Therefore, approximately $17,061.85 must be deposited today.
4. **Calculate the interest earned:**
* Interest = FV - PV
* Interest = $50,000 - $17,061.85
* Interest = $32,938.15 (approximately)
Therefore, approximately $32,938.15 in interest will be earned.
**Question 3.1: Present Value of an Annuity (Monthly Payments, 12% Interest)**
1. **Identify the variables:**
* Payment (PMT): $2,500
* Number of payments (N): 60
* Interest rate per year (r): 12% or 0.12
* Compounding frequency (n): Monthly, so 12 times per year
* Interest rate per period (i): r/n = 0.12 / 12 = 0.01
2. **Use the present value of an ordinary annuity formula:**
* PV = PMT * [(1 - (1 + i)^-N) / i]
3. **Plug in the values:**
* PV = $2,500 * [(1 - (1 + 0.01)^-60) / 0.01]
* PV = $2,500 * [(1 - (1.01)^-60) / 0.01]
* PV = $2,500 * [(1 - 0.5504495) / 0.01]
* PV = $2,500 * [0.4495505 / 0.01]
* PV = $2,500 * 44.95505
* PV = $112,387.63 (approximately)
Therefore, the present value is approximately $112,387.63.
**Question 3.2: Present Value of an Annuity (Monthly Payments, 18% Interest)**
1. **Identify the variables:**
* Payment (PMT): $5,000
* Number of payments (N): 36
* Interest rate per year (r): 18% or 0.18
* Compounding frequency (n): Monthly, so 12 times per year
* Interest rate per period (i): r/n = 0.18 / 12 = 0.015
2. **Use the present value of an ordinary annuity formula:**
* PV = PMT * [(1 - (1 + i)^-N) / i]
3. **Plug in the values:**
* PV = $5,000 * [(1 - (1 + 0.015)^-36) / 0.015]
* PV = $5,000 * [(1 - (1.015)^-36) / 0.015]
* PV = $5,000 * [(1 - 0.580083) / 0.015]
* PV = $5,000 * [0.419917 / 0.015]
* PV = $5,000 * 27.99446
* PV = $139,972.30 (approximately)
Therefore, the present value is approximately $139,972.30.
**Question 4: Life Insurance Policy (Annuity)**
1. **Identify the variables:**
* Payment (PMT): $50,000
* Number of payments (N): 20
* Interest rate per year (r): 8% or 0.08
* Compounding frequency (n): Annually, so 1 time per year
* Interest rate per period (i): r/n = 0.08 / 1 = 0.08
2. **Use the present value of an ordinary annuity formula:**
* PV = PMT * [(1 - (1 + i)^-N) / i]
3. **Plug in the values:**
* PV = $50,000 * [(1 - (1 + 0.08)^-20) / 0.08]
* PV = $50,000 * [(1 - (1.08)^-20) / 0.08]
* PV = $50,000 * [(1 - 0.2145482) / 0.08]
* PV = $50,000 * [0.7854518 / 0.08]
* PV = $50,000 * 9.8181475
* PV = $490,907.38 (approximately)
(a) Therefore, the amount of insurance should be approximately $490,907.38.
(b) **Calculate the total payments:**
* Total payments = PMT * N = $50,000 * 20 = $1,000,000
(c) **Calculate the total interest earned:**
* Total interest = Total payments - PV
* Total interest = $1,000,000 - $490,907.38
* Total interest = $509,092.62 (approximately)
Therefore, the interest earned will be approximately $509,092.62.

Question 1173638: A national census is mandated by the Constitution of Pakistan to be held every ten years. After
the independence of Pakistan in 1947, the first census took place in 1951. Figure 1 reflects the
increase in population of Pakistan during last four national census. The increase between 1972 and
2017 could be approximated by a linear function. If in 1972 the population was 65.309 million
and as per the census data of 2017 the population was 207.774 million;
a) Determine the estimation function P = f (t), where P equals the population of Pakistan and t
equals time measured in years since 1972.
b) Interpret the slope and P intercept
c) According to this function, what will be the estimated population of Pakistan in 2027 census?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this population estimation problem.
**a) Determine the Estimation Function P = f(t)**
1. **Define the Points:**
* (t1, P1) = (0, 65.309 million) - 1972 is the starting year, so t=0.
* (t2, P2) = (45, 207.774 million) - 2017 is 45 years after 1972.
2. **Calculate the Slope (m):**
* m = (P2 - P1) / (t2 - t1)
* m = (207.774 - 65.309) / (45 - 0)
* m = 142.465 / 45
* m ≈ 3.165888888888889 (approximately 3.166 million per year)
3. **Determine the P-intercept (b):**
* Since we're using 1972 as t=0, the P-intercept is the population in 1972, which is 65.309 million.
4. **Write the Linear Equation (P = mt + b):**
* P = 3.166t + 65.309
Therefore, the estimation function is P = 3.166t + 65.309.
**b) Interpret the Slope and P-intercept**
* **Slope (3.166):** The slope represents the average annual increase in Pakistan's population (in millions) between 1972 and 2017. In this case, the population increased by approximately 3.166 million people per year.
* **P-intercept (65.309):** The P-intercept represents the estimated population of Pakistan in 1972, which was 65.309 million.
**c) Estimated Population in 2027 Census**
1. **Calculate the Time (t):**
* 2027 - 1972 = 55 years
2. **Plug t into the Equation:**
* P = 3.166 * 55 + 65.309
* P = 174.13 + 65.309
* P = 239.439 million
Therefore, according to this linear function, the estimated population of Pakistan in 2027 would be approximately 239.439 million.

Question 1176925: Choose the best answer..
1.) Which statement has the same truth value in the statement,
“If the number is even, then it is divisible by two”?
a. If the number is divisible by two, then it is even.
b. If the number is not even, then it is not divisible by two.
c. If the number is not divisible by two, then it is not even.
2.) Given the statement, “If a polygon is pentagon, then it has five
sides”, what is the contrapositive?
a. If a polygon has five sides, then it is a pentagon.
b. If a polygon is not pentagon, then it has no five sides.
c. If a polygon has no five sides, then it is not a pentagon.
For items 3 -5, if p: A = πr2 and q: r = 10
3.) What statement is equivalent to q → p?
a. If r = 10, then A = πr2
b. If A ≠ πr2, then r ≠ 10
c. If r ≠ 10, then A ≠ πr2
4.) What statement is equivalent to ~q → ~p?
a. If r = 10, then A = πr2
b. If A ≠ πr2, then r ≠ 10
c. If r ≠ 10, then A ≠ πr2
5.) What statement is equivalent to ~p → ~q?
a. If r = 10, then A = πr2
b. If A ≠ πr2, then r ≠ 10
c. If r ≠ 10, then A ≠ πr2
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down each question:
**1.) Which statement has the same truth value in the statement, “If the number is even, then it is divisible by two”?**
* **Original statement:** If P, then Q (P: number is even, Q: number is divisible by two)
* We're looking for the **converse**, **inverse**, or **contrapositive** that has the same truth value as the original statement.
* The converse (Q -> P) is: If the number is divisible by two, then it is even.
* The inverse (~P -> ~Q) is: If the number is not even, then it is not divisible by two.
* The contrapositive (~Q -> ~P) is: If the number is not divisible by two, then it is not even.
* The original statement is true.
* The converse is true.
* The inverse is true.
* The contrapositive is true.
* The contrapositive always has the same truth value of the original statement.
* **Answer: c. If the number is not divisible by two, then it is not even.**
**2.) Given the statement, “If a polygon is pentagon, then it has five sides”, what is the contrapositive?**
* **Original statement:** If P, then Q (P: polygon is pentagon, Q: polygon has five sides)
* **Contrapositive:** If ~Q, then ~P (If not Q, then not P)
* **Answer: c. If a polygon has no five sides, then it is not a pentagon.**
**For items 3 -5, if p: A = πr2 and q: r = 10**
**3.) What statement is equivalent to q → p?**
* **q → p:** If q, then p (If r = 10, then A = πr2)
* **Answer: a. If r = 10, then A = πr2**
**4.) What statement is equivalent to ~q → ~p?**
* **~q → ~p:** If not q, then not p (If r ≠ 10, then A ≠ πr2)
* **Answer: c. If r ≠ 10, then A ≠ πr2**
**5.) What statement is equivalent to ~p → ~q?**
* **~p → ~q:** If not p, then not q (If A ≠ πr2, then r ≠ 10)
* **Answer: b. If A ≠ πr2, then r ≠ 10**

Question 1179262: Ten points in the plane are given, with no three collinear. Four
distinct segments joining pairs of these points are chosen at random, all such segments
being equally likely. The probability that some three of the segments form
a triangle whose vertices are among the ten given points is m/n, where m and n
are relatively prime positive integers. Find m + n.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step:
**1. Total Number of Segments:**
* With 10 points, the number of segments that can be formed is given by the combination formula:
* ¹⁰C₂ = 10! / (2! * 8!) = (10 * 9) / 2 = 45
**2. Total Number of Ways to Choose 4 Segments:**
* The number of ways to choose 4 segments from the 45 available is:
* ⁴⁵C₄ = 45! / (4! * 41!) = (45 * 44 * 43 * 42) / (4 * 3 * 2 * 1) = 148,995
**3. Number of Ways to Choose 4 Segments That DO NOT Form a Triangle:**
To find the probability of forming a triangle, it's easier to find the probability of *not* forming a triangle and subtract that from 1.
For 4 segments not to form a triangle, we need to consider the following cases:
* **Case 1: No intersections.** All four segments are completely separate.
* **Case 2: One intersection.** Two segments intersect at a point, but no triangle is formed.
* **Case 3: Two intersections.** Two sets of intersecting segments, but no triangle is formed.
* **Case 4: All 4 segments form a quadrilateral.**
We will use the complementary counting method.
Consider the cases where we *cannot* form a triangle:
* **Case 1: All 4 segments are disjoint.** This is difficult to calculate directly.
* **Case 2: Choose 4 segments such that no 3 form a triangle.**
Instead, let's look at the complementary case:
* **Case 1: 3 of the 4 segments form a triangle.**
* Choose 3 points out of 10 to form a triangle: ¹⁰C₃ = 120
* Choose 1 remaining segment: 42 segments can be chosen that don't form a triangle with the previous 3.
* So, 120 * 42 = 5040 ways to have 3 segments form a triangle, but we overcount, so we must divide by the number of times we can select the same triangle.
* **Case 2: 4 segments form a complete quadrilateral.**
* Choose 4 points out of 10: ¹⁰C₄ = 210
* Each set of 4 points forms 3 possible quadrilaterals.
* 210 * 3 = 630 ways to have 4 segments form a quadrilateral.
Let's use a different approach.
We need to subtract the cases where we cannot form a triangle.
* **Case 1: 4 disjoint segments.**
* **Case 2: 2 pairs of disjoint segments.**
* **Case 3: A "path" of 4 segments.**
* **Case 4: A "star" with 4 segments.**
Instead, let's find the number of ways to form a triangle.
* Choose 3 points out of 10: ¹⁰C₃ = 120
* Choose 1 more segment from the remaining 42 segments: 42
* Total ways to have a triangle: 120 * 42 = 5040
However, we are overcounting.
We can select the 3 segments of the triangle from the chosen 4 segments in 4 ways.
So we have to divide by 4.
5040/4 = 1260
So there are at least 1260 ways.
Probability = 1260/148995 = 84/9933 = 28/3311.
m=28, n=3311
m+n = 3339
**Final Answer:**
m + n = 28 + 3311 = 3339

Question 1180513: Nungesser Corporation’s outstanding bonds have a $1,000 par value, a 9% semiannual coupon, 8 years to maturity, and an 8.5%
YTM. What is the bond’s price?

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to calculate the bond's price:
**1. Determine the Cash Flows:**
* **Coupon Payments:** The bond pays a 9% semiannual coupon, meaning it pays 4.5% of the par value every six months. In dollar terms, this is $1,000 * 0.045 = $45 every six months.
* **Face Value:** At maturity (8 years), the bondholder receives the par value of $1,000.
**2. Calculate the Present Value of the Cash Flows:**
We need to discount each of these cash flows back to their present value using the yield to maturity (YTM) as the discount rate. Since the coupon payments are semiannual, we need to use a semiannual discount rate and number of periods.
* **Semiannual YTM:** 8.5% per year / 2 = 4.25% per period
* **Number of Periods:** 8 years * 2 = 16 periods
Here's a breakdown of the present value calculation:
* **Present Value of Coupon Payments:** This is an annuity. We can use the present value of an annuity formula or discount each payment individually.
* **Present Value of Face Value:** This is a single sum received at maturity. We discount it back 16 periods.
**3. Use a Financial Calculator or Spreadsheet:**
The easiest way to do this is with a financial calculator or spreadsheet software like Excel. Here's how you might input the values:
* **N (Number of Periods):** 16
* **I/YR (Interest Rate per Year):** 4.25 (semiannual YTM)
* **PMT (Payment):** 45 (coupon payment)
* **FV (Future Value):** 1000 (par value)
* **CPT PV (Compute Present Value):** This will give you the bond price.
**4. Result:**
The bond's price will be approximately $1043.30. This means that an investor would be willing to pay $1043.30 for this bond today, given its coupon payments and yield to maturity.

Question 1182658: In a meadow filled with wildflowers, a hiker observes 450 butterflies. Scientists have been studying this meadow for years and have discovered that the number of butterflies depends on the number of flowers in the field. The line of best-fit that they calculated is shown below. If the hiker's observation is analyzed, the residual for the number of butterflies is -27. How many flowers are there in the field?
Number of butterflies = 86+0.25(number of flowers
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem:
1. **Let 'f' represent the number of flowers.** The equation given is:
Number of butterflies = 86 + 0.25f
2. **The hiker observed 450 butterflies.** This is the *actual* (observed) number of butterflies.
3. **The residual is -27.** The residual is the difference between the actual value and the predicted value from the line of best fit. So:
Residual = Actual number of butterflies - Predicted number of butterflies
-27 = 450 - (86 + 0.25f)
4. **Solve for 'f' (number of flowers):**
-27 = 450 - 86 - 0.25f
-27 = 364 - 0.25f
0.25f = 364 + 27
0.25f = 391
f = 391 / 0.25
f = 1564
Therefore, there are **1564** flowers in the field.

Question 1184075: Suppose x=e^(2t).
a)Find the value of the expression x′′′−6x′′+12x′−8x in terms of the variable t. (Enter the terms in the order given.)
My answer is 8e^(2t)-24e^(2t)+24e^(2t)-8e^(2t), correct.
b)Simplify your answer to the previous part and enter a differential equation in terms of the dependent variable x satisfied by x=e^(2t). Enter the derivatives of x using prime notation (x′,x′′,x′′′).
My answer is 8e^(2t)-24e^(2t)+24e^(2t)-8e^(2t)=0 , wrong
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**a) Finding the expression in terms of *t***
You're absolutely correct! Here's how we get there:
* **x = e^(2t)**
* **x' = 2e^(2t)** (First derivative)
* **x'' = 4e^(2t)** (Second derivative)
* **x''' = 8e^(2t)** (Third derivative)
Now, substitute these into the expression x''' - 6x'' + 12x' - 8x:
8e^(2t) - 6(4e^(2t)) + 12(2e^(2t)) - 8(e^(2t)) = 8e^(2t) - 24e^(2t) + 24e^(2t) - 8e^(2t)
**b) Simplifying and the differential equation**
As you've already noticed, the expression simplifies to 0:
8e^(2t) - 24e^(2t) + 24e^(2t) - 8e^(2t) = 0
Therefore, the differential equation satisfied by x = e^(2t) is:
x''' - 6x'' + 12x' - 8x = 0

Question 1209646: Find all solutions $x$ to the equation
5(2x + 1) + 5*2 = 5*x + 5(x + 3).

Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
Find all solutions $x$ to the equation
5(2x + 1) + 5*2 = 5*x + 5(x + 3).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Simplify, step by step 5(2x + 1) + 5*2 = 5*x + 5(x + 3) 10x + 5 + 10 = 5x + 5x + 15 10x + 15 = 10x + 15. At this point, you see that the left side is identical to the right side. It means that there are infinitely many solutions: every real number x is the solution.

Solved, with comprehensive explanations.

Question 1185451: Consider the following statements
X: All policemen wear uniform;
Y: Civil servants do not wear uniform.
If P={policemen}, T={people who wear uniform} and C={civil servants},
(I). draw a Venn diagram to illustrate X and Y
(II). use the diagram to determine which of the following implications are valid or not valid conclusions from X and Y;
(a) Adu wears uniform ____ Adu is a policeman
(b). Ofei is a policeman_____ Ofei is not a Civil Servants.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to analyze the given statements and draw the Venn diagram:
**(I) Venn Diagram:**
The statements can be represented using a Venn diagram with three overlapping sets: P (Policemen), T (People who wear uniform), and C (Civil Servants).
* **Statement X: All policemen wear a uniform.** This means the entire set P is contained within set T.
* **Statement Y: Civil servants do not wear a uniform.** This means sets C and T have no intersection. They are completely separate.
Here's a visual representation:
```
T (Uniform)
-----------------
| |
| P |
| (Policemen) |
| ------ |
| | | |
| | | |
| | | |
| ------ |
| |
| C |
| (Civil Servants)|
|-----------------|
```
**(II) Valid or Not Valid Conclusions:**
**(a) Adu wears uniform ____ Adu is a policeman:**
* **Not Valid.** Adu wearing a uniform (Adu ∈ T) does *not* necessarily mean Adu is a policeman. The Venn diagram shows that the set T (uniform wearers) is larger than the set P (policemen). Adu could be in the part of T that is *outside* of P. There are many people who wear uniforms who are not policemen.
**(b) Ofei is a policeman _____ Ofei is not a civil servant:**
* **Valid.** Since all policemen wear uniforms (X) and no civil servants wear uniforms (Y), it logically follows that no policeman can be a civil servant. If Ofei is a policeman (Ofei ∈ P), then Ofei must also be in T (Ofei ∈ T). Since C and T do not intersect, Ofei cannot be in C (Ofei ∉ C).
**In summary:**
(a) Adu wears uniform **does not imply** Adu is a policeman.
(b) Ofei is a policeman **implies** Ofei is not a civil servant.

Question 1209559: you download game coins for $1 per pack (x) and spent a total of $15
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060)   (Show Source):
You can put this solution on YOUR website!

Sounds to me like you're downloading 15 packs of game coins since they're a dollar a pack and you're spending 15 dollars. Edwin

Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
you download game coins for $1 per pack (x) and spent a total of $15.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Soup of words with no sense.

As far from to be a true Math problem as the ground from heavens.

Question 1186438: 𝑠𝑒𝑛𝑥𝑒cos𝑥𝑑𝑥= Por metodo de sustitucion
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Para resolver la integral ∫sen(x)cos(x)dx por el método de sustitución, podemos seguir estos pasos:
**1. Elegir una sustitución:**
La clave aquí es elegir una sustitución que simplifique la integral. En este caso, una buena opción es:
u = sen(x)
**2. Calcular el diferencial de u:**
Derivamos u con respecto a x:
du/dx = cos(x)
Despejamos du:
du = cos(x)dx
**3. Sustituir en la integral:**
Reemplazamos sen(x) por u y cos(x)dx por du en la integral original:
∫sen(x)cos(x)dx = ∫u du
**4. Integrar con respecto a u:**
La integral de u con respecto a u es simplemente:
∫u du = (u^2)/2 + C
Donde C es la constante de integración.
**5. Deshacer la sustitución:**
Reemplazamos u por sen(x) para volver a la variable original x:
(u^2)/2 + C = (sen(x)^2)/2 + C
**Resultado:**
Por lo tanto, la integral de sen(x)cos(x)dx es:
∫sen(x)cos(x)dx = (sen(x)^2)/2 + C
**Observación:**
También se puede resolver esta integral utilizando la identidad trigonométrica sen(2x) = 2sen(x)cos(x). En ese caso, la integral sería:
∫sen(x)cos(x)dx = (1/2)∫sen(2x)dx = -(1/4)cos(2x) + C
Ambas soluciones son equivalentes, ya que se diferencian solo en la constante de integración.

Question 1209543: Five years ago a man was five times as old as his son.ln 12 years time,the sum of their ages will be 84.Find their present ages.
Found 3 solutions by ikleyn, josgarithmetic, timofer:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.

In 12 years time, the sum of their ages will be 84 ----> Hence, right now the sum of their ages is 84 - 12 - 12 = 84 - 24 = 60 years Right now the sum of their ages is 60 years ---> Hence, five years ago, the sum of their ages was 60 - 5 - 5 = 50. So, according to the problem, five years ago, the father was five times as old as the son and the sum of their ages was 50. But there are no such integer number that satisfy this condition - so the problem is DEFECTIVE.

                    A CONCLUSION and the DIAGNOSIS:

        as presented in the post, the problem is FATALLY DEFECTIVE.

Answer by josgarithmetic(39620)   (Show Source):
You can put this solution on YOUR website!
x, the son's age
y, the father's age

Five years ago a man was five times as old as his son.ln 12 years time,the sum of their ages will be 84.
Find their present ages.
----------
d years ago a man was k times as old as his son. ln p years time,the sum of their ages will be T.
Find their present ages.
-----------

and

and
and if substitute, then

, just in general for the son's present age. But does this need
to be a positive whole number?

In the example given,

, the son's age. If to the nearest month is acceptable, then this should be fine.

If all that is fine then

In case you can not accept non-whole number values, then something is wrong in the exercise description.

Answer by timofer(105)   (Show Source):
You can put this solution on YOUR website!
m for the man and s for his son

and

Some steps
and

and then

not a whole number

You want to recheck the original problem.

Question 1209535: This is surds, rationalisation .
1/√3 +1 - √3 -1/2√3
Answer is 2√3 -3/3
All fractions.
Thank you
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to rationalize and simplify the expression:
1/√3 + 1 - (√3 - 1) / (2√3)
1. **Rationalize the first term:**
Multiply the numerator and denominator by the conjugate of the denominator:
(1 / (√3 + 1)) * ((√3 - 1) / (√3 - 1)) = (√3 - 1) / (3 - 1) = (√3 - 1) / 2
2. **Rationalize the second term:**
Multiply the numerator and denominator by √3:
((√3 - 1) / (2√3)) * (√3 / √3) = (3 - √3) / 6
3. **Combine the terms:**
(√3 - 1) / 2 - (3 - √3) / 6
4. **Find a common denominator (6):**
(3(√3 - 1)) / 6 - (3 - √3) / 6 = (3√3 - 3 - 3 + √3) / 6 = (4√3 - 6) / 6
5. **Simplify:**
(4√3 - 6) / 6 = (2√3 - 3) / 3
So, the simplified and rationalized form of the expression is (2√3 - 3) / 3.

Question 1190765: If f(x)=1/³√2x+1 find f(x+h)-f(x)/h at h=0
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
You're asking for the derivative of f(x) with respect to x, which is the limit of [f(x+h) - f(x)] / h as h approaches 0. Since you're asking to evaluate it *at* h=0, you're effectively asking for the derivative.
Here's how to find f'(x):
1. **Rewrite f(x):** It's often easier to work with fractional exponents: f(x) = (2x + 1)^(-1/3)
2. **Apply the chain rule:**
* The outer function is u^(-1/3), where u = 2x + 1. The derivative of this is (-1/3)u^(-4/3).
* The inner function is u = 2x + 1. The derivative of this is 2.
3. **Combine the derivatives:**
f'(x) = (-1/3)(2x + 1)^(-4/3) * 2
f'(x) = (-2/3)(2x + 1)^(-4/3)
4. **Rewrite with a radical (optional):**
f'(x) = -2 / (3 * ³√((2x + 1)^4)) or f'(x) = -2 / (3(2x+1) * ³√(2x+1))
Since the question asks to evaluate at h=0, but the expression contains only x, we presume that it is asking for the derivative of f(x) with respect to x. Therefore, the answer is:
f'(x) = (-2/3)(2x + 1)^(-4/3) or f'(x) = -2 / (3(2x+1) * ³√(2x+1))

Question 1192189: The following table shows the amount spent by four U.S. airlines to fly one available seat 1 mile in the second quarter of 2014.† Set up a system and then solve using technology. HINT [See the technology note accompanying Example 1.]
Airline United Continental American JetBlue Southwest
Cost (¢) 14.9 14.6 11.9 12.4
Suppose that, on a 3,000-mile New York to Los Angeles flight, United Continental, American, and Southwest flew a total of 235 empty seats, costing them a total of $99,060. If United Continental had three times as many empty seats as American, how many empty seats did each of these three airlines carry on its flight?
+ United Continental _______ empty seats
+ American ______ empty seats
+ Southwest _______ empty seats

Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.

        As the problem is solved in the post by @CPhill, it is one of possible ways/methods,
        but there is another, much more effective and more impressive way to solve the problem
        using only one equation from the very beginning, instead of three.

        Also, it is worth to notice one very important fact:
        in the problem the information about 4 companies is given, but actually only three
        companies participate in the game. The fourth company, Jet Blue, is introduced
        and does present only to confuse the reader and to distract attention.

Let x be the number of empty seats for American. Then the number of empty seats for United Continental is 3x, and the number of empty seats for Southwest is (235 - x - 3x) = 235 - 4x. Now write the total cost equation for empty seats United Cont. American Southwest 3000*0.149*(3x) + 3000*0.146*x + 3000*0.124*(235-4x) = 99060 dollars. Simplify and find x 1341x + 438x + 87420 - 1488x = 99060 291x = 99060 - 87420 = 11640. x = 11640/191 = 40. So, the number of empty seats for American is 40; for United Continental 3*40 = 120, and for Southwest 235-4*40 = 75.

Solved.

Notice that in this solution the system of three equations is not even mentioned.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem:
**1. Set up the variables:**
* Let *u* represent the number of empty seats for United Continental.
* Let *a* represent the number of empty seats for American.
* Let *s* represent the number of empty seats for Southwest.
**2. Set up the equations:**
* **Equation 1 (Total empty seats):** u + a + s = 235
* **Equation 2 (United Continental empty seats):** u = 3a
* **Equation 3 (Total cost of empty seats):** 0.149 * 3000 * u + 0.146 * 3000 * a + 0.124 * 3000 * s = 99060
**3. Simplify Equation 3:**
447u + 438a + 372s = 99060
**4. Substitute Equation 2 into Equation 1 and Equation 3:**
* **New Equation 1:** 3a + a + s = 235 => 4a + s = 235
* **New Equation 3:** 447(3a) + 438a + 372s = 99060 => 1341a + 438a + 372s = 99060 => 1779a + 372s = 99060
**5. Solve the system of two equations (New Equation 1 and New Equation 3):**
One way to do this is using substitution or elimination. Let's use substitution:
* From New Equation 1: s = 235 - 4a
Substitute this into New Equation 3:
1779a + 372(235 - 4a) = 99060
1779a + 87420 - 1488a = 99060
291a = 11640
a = 40
Now substitute a = 40 back into the equation for s:
s = 235 - 4(40)
s = 235 - 160
s = 75
Finally, substitute a = 40 back into the equation for u:
u = 3(40)
u = 120
**Solution:**
* United Continental: 120 empty seats
* American: 40 empty seats
* Southwest: 75 empty seats

Question 1209496: Express x^2 + 6kx + 144 in the form of (x+p)^2 + q.
Find the range of values of k suck that the x^2 + 6kx + 144 is positive for all values of x.
[the answer for (a) is solved already, I need the second part please.]
[the answer for (b) is -4 Thank you!
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
(a) Express x^2 + 6kx + 144 in the form of (x+p)^2 + q.
(b) Find the range of values of k suck that the x^2 + 6kx + 144 is positive for all values of x.
[the answer for (a) is solved already, I need the second part please.]
[the answer for (b) is -4 Thank you!
~~~~~~~~~~~~~~~~~~~~~~~~

I will work for part (b), ONLY.

Consider this polynomial x^2 + 6kx + 144 and find the discriminant for it d = b^2 - 4ac = (6k)^2 - 4*1*144 = = 36k^2 - 4*144. The polynomial x^2 + 6kx + 144 is positive for all real values of x if and only if the discriminant is negative D < 0 36k^2 - 4*144 < 0, or 36k^2 < 4*144, or k^2 < = = 4. Taking the square root of both sides, we get |k| < 4, or, which is the same, -4 < k < 4. ANSWER. The range for k is -4 < k < 4.

Solved.

Question 1209488: Find the range of values of k for which the expression x^2 + kx + (k+3) is positive for all real values of x. Hence, fine the range of values of x for which (2x-1)(3-x)/x^2+4x+7 < 0
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
(a) Find the range of values of k for which the expression x^2 + kx + (k+3) is positive for all real values of x.
(b) Hence, find the range of values of x for which (2x-1)(3-x)/(x^2+4x+7) < 0
~~~~~~~~~~~~~~~~~~~~~~~

        In this assignment, there are two tasks: (a) and (b).
        I will solve them separately to avoid mess.

        Also notice that I edited your inequality in part (b) according to common sense.

Part (a) Consider the discriminant of the quadratic polynomial x^2 + kx + (k+3). The discriminant is d = b^2 - 4ac = k^2 - 4*(k+3) = k^2 - 4k - 12 = (k-6)*(k+2). We see that the discriminant is negative in the interval -2 < k < 6. It means that the polynomial does not have real zeroes if -2 < k < 6. From the other side hand, its leading coefficient at x^2, "1", is positive. It means that the polynomial x^2 + kx + (k+3) is always positive, for all real values of x, if k is in the open interval (-2,6). Thus, part (a) is solved/answered completely. Part (b) Consider this rational function . Its denominator is of the form x^2 + kx + (k+3) at k = 4. We considered such polynomials in part (a) and proved that for k from the interval (-2,6) the polynomial is always positive, for all real values of x. The value of k= 4 is from this interval - so, the polynomial x^2 + 4x + 7 in the denominator is always positive, for all real values of x. Therefore, inequality < 0 (1) is equivalent to this simplified inequality (2x-1)*(3-x) < 0. (2) This inequality (2) has the leading coefficient -2 at x^2 and the roots 1/2 and 3, so the left side is the downward parabola with x-intercepts 1/2 and 3. So, the inequality (2) has the solution set < x < 3. It implies that inequality (1) has the same solution set < x < 3. Thus the range of values of x for which < 0 is (,). ANSWER At this point, part (b) is solved completely.

The solution is complete and all questions are answered.

Question 1209490: Show that x^2-x+1 is always positive for all real values of x. Hence, or otherwise, find the range of values of ‘a’ if the inequality x^2+ax-2/x^2-x+1 < 2 is satisfied for all real values of x.
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website!
.
(a) Show that x^2-x+1 is always positive for all real values of x.
(b) Hence, or otherwise, find the range of values of ‘a’ if the inequality x^2+ax-2/x^2-x+1 < 2
is satisfied for all real values of x.
~~~~~~~~~~~~~~~~~~~~~~~

        In this assignment, there are two tasks: (a) and (b).
        I will solve them separately to avoid mess.

Part (a) Consider the discriminant of the quadratic polynomial x^2 - x + 1. The discriminant is d = b^2 - 4ac = (-1)^2 - 4*1*1 = 1 - 4 = -3. We see that the discriminant is negative. It means that the polynomial does not have real zeroes. From the other side hand, its leading coefficient at x^2, "1", is positive. It means that the polynomial x^2 - x + 1 is always positive, for all real values of x. Thus the statement (a) is proved and part (a) is completed. Part (b) Consider this rational function . As we proved in part (a), its denominator is always positive, for all real values of x. Therefore, while solving inequality < 2, (1) we can multiply both its sides by the positive factor x^2 - x + 1, without flipping the inequality sign. By doing it, we get the equivalent inequality x^2 + ax - 2 < 2*(x^2 - x + 1). (2) Simplify it step by step x^2 + ax - 2 < 2x^2 - 2x + 2, 0 < x^2 - (2-a)x + 4, x^2 - (2-a)x + 4 > 0. (3) Thus, inequality (1) is held for all real x if and only if inequality (3) is held for all real x. To find, what it implies for "a", let's consider the discriminant of the quadratic polynomial in the left side of (3). It is D = b^2 - 4ac = (-(2-a))^2 - 4*4 = (2-a)^2 - 16. Inequality (3) is held for all real x if and only if discriminant D is positive D > 0, or (2-a)^2 - 16 >0, or (2-a)^2 > 16, or |2-a| > , or |2-a) > 4. (4) The set of solutions to inequality (4) are all values of "a" that are remoted from 2 farther than 4 units. So, the set of "a" when inequality (1) is held for all real x, is the union (,) U (,). ANSWER. The set of values "a", such that inequality < 2 is satisfied for all real values of x is the union (,) U (,).

At this point, the problem is solved completely for both parts (a) and (b).