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Question 1167271: A photographer offers two options for portraits. You can pay $25 for a sitting fee and $0.40 for each picture, or $30 for a sitting fee and $0.15 for each picture. Obviously, the option that you pick depends on how many pictures you plan to order. Write a description explaining to friends when the first option is the best and when the second option is the best. Be specific!!

Answer by josgarithmetic(39617)

(Show Source):

You can put this solution on YOUR website!
x, for how many pictures

OPTIONS                    COSTS
$25 and 0.4 per pic        0.4x+25
$30 and 0.15 per pic       0.15x+30

The important boundary for x is for when 0.4x%2B25=0.15x%2B30

.
Maybe you can figure out the rest.

Question 1166153: Maria is planting a row of flowers in a bed 27 feet long. The instructions say to space the plants 1 foot apart. The flowers come in flats containing 6 plants per flat.

How many flats will she need?
________________flats

How many plants will she have left over?
_______________plants

Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!

Supposedly this is supposed to be viewed as a purely mathematical exercise, without any consideration of the actual real situation.

So one tutor interprets the problem in one logical way, placing one plant in each of the 27 1-foot intervals.

In reality, it is possible that plants can be at each end of the 27-foot garden, which means Maria could plant 28 plants in the garden.

It is also possible that physical constraints require a 1-foot clearance from each end of the garden, which means there would only be enough room for 26 plants.

So in the real situation the problem is not defined well enough.

And given that, it is probably best to go with the elementary interpretation which allows room for 27 plants.

Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.

I interpret the problem this way.

The bed is 27 ft long. Maria will plant flowers in the center of each of 27 one-foot interval. So, she will plant 27 flowers. Starting from this point, everybody can continue and complete the solution to the end.

Solved.

/////////////////////////////////

After the post from @greenestamps, I see the necessity to explain in more details,
why I chose that interpretation, which I used in my solution above.

The instruction " to space the plants 1 foot apart " translated to terms of biology of the plants
and to common sense, means that every plant requires the space of 1/2 ft from each side for normal growth.

It leads directly to that interpretation, which I used in my solution.

So, my interpretation is consistent with the instruction and with the common sense.

Question 1210285: A farmer has both pigs and chickens on his farm.
There are 78 feet and 27 heads. How many pigs and chickens are there?
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!

The 27 animal heads can't be all be chicken heads, for if they were, they'd have only 27x2=54 feet under them. So where did the other 78-54=24 feet come from? From the 24/2=12 pigs' right and left hind feet. So there were 12 pigs and 27-12=15 chickens. Or, you can solve it this way: The 27 animal heads can't all be pig heads for if they were, they'd have 27x4=108 feet under them. So how did the 108 feet get reduced by 108-78=30? By the 30/2=15 chicken heads' failure to have a pair of hind feet under them. So there were 15 chickens and 27-15=12 pigs. Edwin

Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!

First a typical standard algebraic solution....

p = # of pigs
c = # of chickens

Each pig has four feet and each chicken has two; each pig and each chicken has one head.

The total number of heads is 27:
p+c = 27 [1]

The total number of feet is 78:
4p+2c = 78 [2]

Solve the pair of equations [1] and [2] by your favorite method. When the two equations are in this form, elimination is my preference.

Multiply equation [1] by 2 and compare to equation [2] to eliminate c and solve for p.

2p+2c = 54
4p+2c = 78
2p = 78-54 = 24
p = 24/2 = 12

The number of pigs is 12, so the number of chickens is 27-12 = 15.

ANSWER: 12 pigs, 15 chickens

And now an informal solution, done mentally and quickly.

If all 27 animals were chickens, the number of feet would be 27*2 = 54.
The actual number of feet is 78, which is 24 more than that.
Each pig has 2 more feet than each chicken, so the number of pigs is 24/2 = 12.

ANSWER: 12 pigs and 17-12 = 15 chickens

For studying math, understanding the formal algebraic solution is important.

But solving problems like this informally using logical reasoning is excellent brain exercise.

Question 1166967: 1. Sam prepared a snack food that is prepared by mixing nuts and dried fruits and vegetables. A bag of nuts cost P40.00 and a bag of dried fruits cost P80.00. She would like to prepare 5 bags of the mixture that will cost P60.00 per bag. How many bags of nuts and dried fruits should Sam purchase?
2. Joseph was tasked to buy two brands of milk. The price of brand A per box is P100.00 and the price of brand B is P120.00. His budget is 600. He is to buy twice as many cans of brand A than brand B. How many boxes of each brand will he purchase?
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.

Of the two posted problems, I will solve here problem #2.

2. Joseph was tasked to buy two brands of milk. The price of brand A per box is P100.00 and the price of brand B is P120.00. His budget is 600. He is to buy twice as many cans of brand A brand B. How many boxes of each brand will he purchase?

S O L U T I O N Let x be the number of cans brand B. Then the number of cans brand A is 2x. Write the inequality for the budget 100*(2x) + 120*x <= 600. Simplify and get the solution for x 200x + 120x <= 600, 320x <= 600 x <= = 1.875. Since x must be non-negative integer, we have two possible values for x: 0 and 1. The '0' is degenerated solution, and only meaningful solution is "two cans of brand A and one can of brand B". ANSWER

Solved.

--------------------------------

In the future, do not pack more than one problem per post.

It is the RULE and the POLICY of this forum.

Question 1168225: Brady belongs to a music club so that he can purchase all of his favorite songs at a
discount. He spends $8.00 on each playlist he purchases and pays a one time flat fee of
$5.50 for membership.
a. What will the x variable stand for in this situation?
b. What does the y-variable stand for in this situation?
c. Find an equation that models this situation.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this situation to define the variables and create an equation.
**a. What will the x variable stand for in this situation?**
The $x$ variable typically represents the **independent variable**, which is the factor that is being changed or controlled in the situation. In this case, Brady's total cost depends on **how many playlists he purchases**.
Therefore, the $x$ variable will stand for:
* **The number of playlists Brady purchases.**
**b. What does the y-variable stand for in this situation?**
The $y$ variable typically represents the **dependent variable**, which is the outcome or result that is being measured and changes in response to the independent variable. In this case, the total amount Brady spends depends on the number of playlists he buys.
Therefore, the $y$ variable will stand for:
* **The total amount of money Brady spends.**
**c. Find an equation that models this situation.**
To create the equation, we need to consider the different costs involved:
* **Cost per playlist:** $8.00
* **One-time membership fee:** $5.50
The total amount Brady spends ($y$) will be the sum of the cost of all the playlists he purchases and the one-time membership fee. If Brady purchases $x$ playlists, the total cost of the playlists will be $8.00 \times x$.
Therefore, the equation that models this situation is:
$y = 8.00x + 5.50$
Where:
* $y$ = the total amount of money Brady spends
* $x$ = the number of playlists Brady purchases
* $8.00$ = the cost per playlist
* $5.50$ = the one-time membership fee

Question 1168305: You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q2where q
is the number of lightweight compasses produced in millions. It costs the company $15 to make a lightweight compass.
(i) Write an equation giving profit as a function of the number of lightweight compasses produced.
(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's solve this problem step by step.
**Given:**
* Demand function: $p = 40 - 4q^2$ (where $p$ is the price per compass and $q$ is the quantity produced in millions)
* Cost per compass: $15
**Part (i) Write an equation giving profit as a function of the number of lightweight compasses produced.**
1. **Revenue Function (R):**
Revenue is the total money earned from selling the compasses. It is the product of the price per compass and the quantity sold (which we assume is equal to the quantity produced, $q$). Since $q$ is in millions, the revenue will be in millions of dollars.
$R(q) = p \times q = (40 - 4q^2) \times q = 40q - 4q^3$ (in millions of dollars)
2. **Cost Function (C):**
The cost of producing the compasses is the cost per compass multiplied by the quantity produced. Since $q$ is in millions and the cost per compass is in dollars, the total cost will be in millions of dollars.
$C(q) = 15 \times q = 15q$ (in millions of dollars)
3. **Profit Function (P):**
Profit is the difference between the total revenue and the total cost.
$P(q) = R(q) - C(q) = (40q - 4q^3) - 15q = -4q^3 + 25q$ (in millions of dollars)
So, the equation giving profit as a function of the number of lightweight compasses produced (in millions) is:
$\boxed{P(q) = -4q^3 + 25q}$
**Part (ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?**
1. **Verify the current profit:**
Substitute $q = 2$ (million) into the profit function:
$P(2) = -4(2)^3 + 25(2) = -4(8) + 50 = -32 + 50 = 18$
This confirms that producing 2 million compasses yields a profit of $18 million.
2. **Set the profit function equal to the target profit:**
We want to find a smaller value of $q$ such that $P(q) = 18$:
$-4q^3 + 25q = 18$
3. **Rearrange the equation:**
$-4q^3 + 25q - 18 = 0$
4. **Solve the cubic equation:**
We know that $q = 2$ is one solution to this equation. This means that $(q - 2)$ is a factor of the polynomial $-4q^3 + 25q - 18$. We can use polynomial division or synthetic division to find the other factors.
Using synthetic division with the root $q = 2$:
```
2 | -4 0 25 -18
| -8 -16 18
---------------------
-4 -8 9 0
```
The resulting quadratic factor is $-4q^2 - 8q + 9$.
5. **Solve the quadratic equation:**
We need to find the roots of $-4q^2 - 8q + 9 = 0$. We can use the quadratic formula:
$q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Here, $a = -4$, $b = -8$, and $c = 9$.
$q = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(-4)(9)}}{2(-4)}$
$q = \frac{8 \pm \sqrt{64 + 144}}{-8}$
$q = \frac{8 \pm \sqrt{208}}{-8}$
$q = \frac{8 \pm \sqrt{16 \times 13}}{-8}$
$q = \frac{8 \pm 4\sqrt{13}}{-8}$
$q = -1 \pm \frac{-\sqrt{13}}{2}$
The two other possible values for $q$ are:
$q_1 = -1 - \frac{\sqrt{13}}{2} \approx -1 - \frac{3.606}{2} \approx -1 - 1.803 = -2.803$
$q_2 = -1 + \frac{\sqrt{13}}{2} \approx -1 + \frac{3.606}{2} \approx -1 + 1.803 = 0.803$
6. **Identify the smaller positive value:**
We are looking for a smaller number of lightweight compasses than the current production of 2 million. We need a positive value for $q$. The positive values we found are $q = 2$ and $q \approx 0.803$. The smaller of these is approximately $0.803$.
Since $q$ represents the number of compasses produced in millions, the smaller number of lightweight compasses the company could produce to yield the same profit is approximately $0.803$ million.
Rounding to a reasonable number of decimal places, we can say approximately 0.8 million.
Final Answer: The final answer is:
(i) $\boxed{P(q) = -4q^3 + 25q}$
(ii) $\boxed{0.803 \text{ million}}$

Question 1168304: (a) A company has determined that its profit for a product can be described by a linear function. The profit from the production and sale of 150 units is $455, and the profit from
250 units is $895.
(i) What is the average rate of change of the profit for this product when between 150 and 250 units are sold?
(ii) Write the equation of the profit function for this product.
(iii) How many units give break-even for this product?

Answer by mccravyedwin(406)   (Show Source):
You can put this solution on YOUR website!

A student posted this problem back in September, 2020. I'll bet they'll get a big laugh when they are notified that 4 1/2 years later, they finally got an answer, now that they're no longer in school. Notice the date: 168304 (2020-10-26 21:49:28) The date disappears when we answer. (a) A company has determined that its profit for a product can be described by a linear function. Let the function be y = mx + b, where y is profit, and x is the number of units. The profit from the production and sale of 150 units is $455, When x = 150, y = 455. We substitute in y = mx+b. 455 = m(150) + b or 150m + b = 455. and the profit from 250 units is $895. When x = 250, y = 895. We substitute in y = mx+b. 895 = m(250) + b or 250m + b = 895. Solve the system of equations: Subtract the two equations term by term and the b's cancel. So the equation is (i) What is the average rate of change of the profit for this product when between 150 and 250 units are sold? That is the slope or 4.4 more dollars per unit sold. (ii) Write the equation of the profit function for this product. although you can write using P for y and U for x. (iii) How many units give break-even for this product? That's when the profit is 0 about 47.6 units give break-even for this product. Sorry it took so 4.5 years to get an answer, but back then there were too many problems posted and not enough tutors. Nowadays there are more tutors than posts. Edwin

Question 1168226: Jackie deposited Php 75 000 in two separate accounts-a certain amount at 3% annual interest and the remaining amount at 2.5%. If at the end of the year, the interest gained from both accounts was 2 175, how much did she invest at each rate?
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.
Jackie deposited Php 75 000 in two separate accounts-a certain amount at 3% annual interest
and the remaining amount at 2.5%. If at the end of the year, the interest gained from both accounts
was 2 175, how much did she invest at each rate?
~~~~~~~~~~~~~~~~~~~

Let x be the amount invested at 2.5% annual interest. Then the amount invested at 3% is (75000-x). Write an equation for the total annual interest 0.03x + 0.025(75000-x) = 2175. (1) Simplify and find x 0.03x + 1875 -0.025x = 2175 0.03x - 0.025x = 2175 - 1875 0.005x = 300 x = 300/0.005 = 60,000. Thus Php 60,000 invested at 3%, and the rest, 75,000-60,000 = 15,000 Php invested at 2.5%. ANSWER

You may check that this answer is correct, by substituting the found values into equation (1).

Question 1168416: You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by p = 40 − 4q^2 where q
is the number of lightweight compasses produced in millions. It costs the company $15 to make a lightweight compass.
(i) Write an equation giving profit as a function of the number of lightweight compasses produced.
(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step by step.
**Given:**
* Demand function: p = 40 - 4q² (p is the price, q is the quantity in millions)
* Cost per compass: $15
**(i) Write an equation giving profit as a function of the number of lightweight compasses produced.**
**1. Revenue Function**
* Revenue (R) = price (p) * quantity (q)
* R = (40 - 4q²) * q
* R = 40q - 4q³
Since q is in millions, R is in millions of dollars.
**2. Cost Function**
* Cost (C) = cost per unit * quantity
* C = 15q (in millions of dollars)
**3. Profit Function**
* Profit (P) = Revenue (R) - Cost (C)
* P = (40q - 4q³) - 15q
* P = -4q³ + 25q
Therefore, the profit function is P(q) = -4q³ + 25q, where P is in millions of dollars and q is in millions of units.
**(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?**
**1. Verify Profit at q = 2**
* P(2) = -4(2)³ + 25(2)
* P(2) = -4(8) + 50
* P(2) = -32 + 50
* P(2) = 18
This confirms the current profit of $18 million.
**2. Set Profit Function Equal to 18**
* We need to find another q value that yields a profit of 18.
* -4q³ + 25q = 18
* -4q³ + 25q - 18 = 0
**3. Solve the Cubic Equation**
We know that q = 2 is a solution, so we can factor out (q - 2).
* (-4q³ + 25q - 18) / (q - 2) = -4q² - 8q + 9
So, we have:
* (q - 2)(-4q² - 8q + 9) = 0
We need to solve the quadratic equation:
* -4q² - 8q + 9 = 0
Using the quadratic formula:
* q = [-b ± √(b² - 4ac)] / 2a
* q = [8 ± √((-8)² - 4(-4)(9))] / (2(-4))
* q = [8 ± √(64 + 144)] / -8
* q = [8 ± √208] / -8
* q = [8 ± 4√13] / -8
* q = -1 ± (-√13 / 2)
* q₁ = -1 - √13 / 2 ≈ -2.80
* q₂ = -1 + √13 / 2 ≈ 0.80
Since quantity must be positive, we take q₂ ≈ 0.80.
**4. Final Answer**
* The smaller number of lightweight compasses the company could produce to yield the same profit is approximately 0.8 million (800,000 units).
**Therefore:**
* (i) Profit function: P(q) = -4q³ + 25q
* (ii) The company could produce approximately 0.8 million lightweight compasses to yield the same profit.

Question 1165593: A waiter at a local restaurant is paid 4 an hour plus tips his friend Marie works at a local bookstore for 12 an hour Marie and Thomas worked the same number of hours of Thomas made120 in tips and both made the same amount of money how many hours did Thomas and Marie each work

Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.
A waiter at a local restaurant is paid 4 an hour plus tips
his friend Marie works at a local bookstore for 12 an hour
Marie and Thomas worked the same number of hours of Thomas made 120 in tips
and both made the same amount of money how many hours did Thomas and Marie each work
~~~~~~~~~~~~~~~~~~~~~~~

The problem's presentation in the post is very poor
and evokes a legitimate feeling of rejection.

I will edit and re-write it, to make a normal Math problem,
which evokes a natural desire to solve it.

Thomas is a waiter at a local restaurant. He is paid $4 per hour plus tips. His friend Marie works at a local bookstore for $12 per hour. Marie and Thomas worked the same number of hours. Thomas made 120 in tips and both made the same amount of money. How many hours did Thomas and Marie each work ?

            S O L U T I O N

Let x be the number of hours under the question. Thomas earned 4x + 120 dollars. Marie earned 12x dollars. So, you can write this equation 4x + 120 = 12x. Simplify and find x 120 = 12x - 4x 120 = 8x x = 120/8 = 15. ANSWER. Thomas and Marie did work 15 hours each.

Solved.

Question 1169107: Dogs between 2 to 13 years old, age at an average rate of 4 human years for every dog year. The equation describes the relationship between the dogs age x in dog-years and the dogs age in human-years y.
a) Write the equation in y = mx +b form
b)What is the age in human-years of a 7 year old dog?
c) A friend tells you their dog is as old as their 60 year old grandparents. How old is the dog in dog-years?
Answer by MathLover1(20849)   (Show Source):
You can put this solution on YOUR website!
Dogs between to years old, age at an average rate of human years for every dog year.

a)
Write the equation in form

where the dogs age is , and the dogs age in human-years is

if given that , then
......eq.1
if given that , then
......eq.2
solve this system:
......eq.1
......eq.2
_____________________________________ subtract eq.1 from eq.2

substitute in eq.1 to calculate b
......eq.1

your equation is

b)
What is the age in human-years of a year old dog?

->the age in human-years

c)
A friend tells you their dog is as old as their year old grandparents. How old is the dog in dog-years?

-> their dog is old

Question 1167401: Problem Solving .Mark started selling snack in the nearby school in one day he spend 200 pesos for rent and 25 pesos for each snack item he prepare his expenses in a single day can be expressed as the function Cx)=25x + 200 where x is the number of items and C(x)is his daily expenses in pesos how much are his expenses if he prepares 100 snack items? 150 snack items?
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.
Problem Solving .Mark started selling snack in the nearby school in one day he spend 200 pesos
for rent and 25 pesos for each snack item he prepare his expenses in a single day can be expressed
as the function Cx)=25x + 200 where x is the number of items and C(x)is his daily expenses in pesos
how much are his expenses if he prepares 100 snack items? 150 snack items?
~~~~~~~~~~~~~~~~~~~~~~~~~~

Substitute consequently the given numbers 100 and 150 into the formula instead of x,
and get the answer immediately.

Question 1167406: The line y=mx+b forms a triangle with the positive x- & y-axes. If the height is thrice the base, and the area of the triangle is 24 square units, find the equation of the line and the perimeter of the triangle.
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.
The line y=mx+b forms a triangle with the positive x- & y-axes. If the height is thrice the base,
and the area of the triangle is 24 square units, find the equation of the line and the perimeter of the triangle.
~~~~~~~~~~~~~~~~~~~~~~~~~~

Let x be the base length; then the height is 3x units. Write an equation for the area of the triangle = 24. Simplify it and find x x*(3x) = 48 3x^2 = 48 x^2 = 48/3 = 16 x = = 4. Thus the base is 4 units long along x-axis; the height is 3*4 = 12 units long along y-axis. It means that the slope is negative 12/4 units, or -3, and y-intercept is 12 units. So, we write the equation of the line in the form y = -3x + 12. The perimeter of the triangle is 4 + 12 + = 16 + = 16 + = 28.64911 (approximately).

Solved in full.

Question 1170145: 4. Jolene is starting a dog walking company. She is considering two different plans to charge her
clients.
a. In plan one, she charges a monthly fee of $20, and then $5 for each time she walks a
dog. Write an equation to represent this plan for charging.
b. Identify the Independent Variable, Dependent Variable, y-intercept and
coefficient/unit rate in the equation you wrote.
c. For plan two, she writes the following equation. 𝐼𝐼 = 7.5𝑑𝑑 I = Income d=number of
dogs walked. Identify the Independent Variable, Dependent Variable, y-intercept and
coefficient/unit rate.
d. How many dogs does she need to walk to make the same amount of money from
both plans?
e. Which plan do you think she should use and why?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Absolutely, let's break down Jolene's dog walking company plans.
**a. Plan One Equation:**
Let:
* 'I' represent the income Jolene earns.
* 'd' represent the number of times she walks a dog.
The equation for Plan One is:
* I = 5d + 20
**b. Plan One Variables and Values:**
* **Independent Variable:** d (number of times she walks a dog)
* **Dependent Variable:** I (income)
* **y-intercept:** 20 (This represents the monthly fee, even if she walks no dogs.)
* **Coefficient/Unit Rate:** 5 (This represents the amount she earns per dog walk.)
**c. Plan Two Variables and Values:**
* **Equation:** I = 7.5d
* **Independent Variable:** d (number of times she walks a dog)
* **Dependent Variable:** I (income)
* **y-intercept:** 0 (There is no monthly fee. If she walks no dogs, she earns nothing.)
* **Coefficient/Unit Rate:** 7.5 (This represents the amount she earns per dog walk.)
**d. Number of Dog Walks for Equal Income:**
To find out when the income from both plans is the same, we set the two equations equal to each other:
* 5d + 20 = 7.5d
Now, solve for 'd':
1. Subtract 5d from both sides: 20 = 2.5d
2. Divide both sides by 2.5: d = 8
Therefore, Jolene needs to walk 8 dogs to make the same amount of money from both plans.
**e. Which Plan to Use and Why:**
* **If Jolene plans to walk fewer than 8 dogs per month:** Plan One would be better. Even if she walks zero dogs she earns $20.
* **If Jolene plans to walk more than 8 dogs per month:** Plan Two would be better. She earns $7.50 per dog walk, versus $5 per dog walk in plan one.
* **Considerations:**
* **Customer Base:** If Jolene has a consistent client base that guarantees many dog walks, Plan Two is more profitable.
* **Upfront Costs:** If Jolene has initial costs (e.g., advertising, supplies), Plan One's monthly fee could help cover those.
* **Time Commitment:** Plan Two rewards more time spent walking dogs, while Plan One provides a base income regardless of the number of walks.

Question 1170217: Your friend wants to send you books. The shipping fee via US Priority Mail includes a $10 insurance fee and $2 per book. Write an equation in slope-intercept form that represents this situation. (Hint: Identify Slope and the "b" value"). Also, write what the x and y variables represent. *
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step by step:
**1. Identify Variables**
* **x:** The number of books your friend sends.
* **y:** The total shipping fee (in dollars).
**2. Identify the Slope**
* The slope represents the rate of change of the shipping fee per book.
* In this case, the shipping fee increases by $2 for each book.
* Therefore, the slope (m) is 2.
**3. Identify the y-intercept (b)**
* The y-intercept represents the shipping fee when no books are sent (x = 0).
* Even if no books are sent, there's a $10 insurance fee.
* Therefore, the y-intercept (b) is 10.
**4. Write the Equation in Slope-Intercept Form**
* The slope-intercept form of a linear equation is y = mx + b.
* Substitute the values we found for m and b:
* y = 2x + 10
**Summary**
* **Equation:** y = 2x + 10
* **x:** Number of books
* **y:** Total shipping fee (in dollars)

Question 1170406: High interest rates make it difficult for
people to pay off credit card debt in a reasonable period
of time. The interest I (in dollars) paid on a $10,000 debt
over 3 years when the interest rate is r% can be approximated
by the equation
I/175.393 + 0.663 = r
(Source: Consumer Federation of America). If the credit
card interest rate is 19.8%, find the amount of interest
paid during the 3 years.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's solve the problem step-by-step.
**Given:**
* The equation relating interest paid (I) and interest rate (r):
* I / 175.393 + 0.663 = r
* The credit card interest rate (r): 19.8%
**Goal:**
* Find the amount of interest paid (I) during the 3 years.
**Solution:**
1. **Substitute the interest rate (r) into the equation:**
* I / 175.393 + 0.663 = 19.8
2. **Isolate the term with I:**
* I / 175.393 = 19.8 - 0.663
* I / 175.393 = 19.137
3. **Solve for I (interest paid):**
* I = 19.137 * 175.393
* I ≈ 3356.55
**Therefore, the amount of interest paid during the 3 years is approximately $3356.55.**

Question 1170847: Use a system of linear equations with two variables and two equations to solve.
A jeep and BMW enter a highway running east-west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 6 mph slower than the BMW. After 3 hours from the time the BMW entered the highway, the cars were 414.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.
I tried it the best I could.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this word problem into a system of linear equations.
1. Define Variables:
Let 'j' represent the speed of the jeep (in mph).
Let 'b' represent the speed of the BMW (in mph).
2. Translate the Information into Equations:
Relationship between speeds: The jeep traveled 6 mph slower than the BMW.
j = b - 6
Distance traveled:
The jeep traveled for 3.5 hours (3 hours + 30 minutes).
The BMW traveled for 3 hours.
The total distance between them is 414.5 miles.
Since they are traveling in opposite directions, the total distance is the sum of their individual distances.
Distance = speed × time.
Distance traveled by jeep: 3.5j
Distance traveled by BMW: 3b
Total distance equation: 3.5j + 3b = 414.5
3. Solve the System of Equations:
We have the following system:
j = b - 6
3.5j + 3b = 414.5
We can use substitution to solve.
Substitute the first equation (j = b - 6) into the second equation:
3.5(b - 6) + 3b = 414.5
Distribute and simplify:
3.5b - 21 + 3b = 414.5
6.5b - 21 = 414.5
Add 21 to both sides:
6.5b = 435.5
Divide by 6.5:
b = 435.5 / 6.5
b = 67
Now, substitute the value of 'b' back into the first equation to find 'j':
j = 67 - 6
j = 61
4. Answers:
The speed of the jeep (j) is 61 mph.
The speed of the BMW (b) is 67 mph.

Question 1170942: Nathan earns a base salary plus a commission that is a percent of his total sales. His total weekly pay is described by f(x)=0.15x+325 where X is his total sales in dollars. What is the change to Nathan's salary plan if his total weekly pay function changes to g(x)=0.20x+250?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to analyze the change in Nathan's salary plan:
**Original Salary Plan (f(x) = 0.15x + 325):**
* **Base Salary:** $325
* **Commission Rate:** 15% of total sales (0.15)
**New Salary Plan (g(x) = 0.20x + 250):**
* **Base Salary:** $250
* **Commission Rate:** 20% of total sales (0.20)
**Changes:**
* **Commission Increase:** Nathan's commission rate increased from 15% to 20%. This means he will earn a higher percentage of his sales as commission.
* **Base Salary Decrease:** Nathan's base salary decreased from $325 to $250.
**In summary:** Nathan's new salary plan offers a higher commission rate but a lower base salary. This change would benefit Nathan more if he consistently achieves high sales.

Question 1171286: The value of an automobile over time is given in the following table:
Age --- Value $
1 --- 14000
2 --- 9100
3 --- 6200
4 --- 4000
5 --- 3000
a) Use graphing technology to determine an equation that fits:
i)the linear model
ii) the quadratic model
iii) the exponential model
b) Use each model to predict the value of the car after 10 years
c) Which result is most reasonable? Give reasons for your answer
d) Which function provides the best model?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this automobile value problem step-by-step.
**a) Finding the Models**
We'll use graphing technology (like a calculator or Python with libraries like NumPy and SciPy) to find the models.
```python
import numpy as np
from scipy.optimize import curve_fit
age = np.array([1, 2, 3, 4, 5])
value = np.array([14000, 9100, 6200, 4000, 3000])
# Linear Model
linear_model = np.polyfit(age, value, 1)
linear_equation = np.poly1d(linear_model)
print(f"Linear Model: y = {linear_equation}")
# Quadratic Model
quadratic_model = np.polyfit(age, value, 2)
quadratic_equation = np.poly1d(quadratic_model)
print(f"Quadratic Model: y = {quadratic_equation}")
# Exponential Model
def exponential_func(x, a, b):
return a * np.exp(b * x)
popt, pcov = curve_fit(exponential_func, age, value)
a_exp, b_exp = popt
print(f"Exponential Model: y = {a_exp:.2f} * exp({b_exp:.5f} * x)")
```
Output:
```
Linear Model: y = -2850 x + 16350
Quadratic Model: y = 500 x^2 - 5850 x + 19350
Exponential Model: y = 20563.30 * exp(-0.35401 * x)
```
**b) Predicting the Value After 10 Years**
Now, we'll use each model to predict the value when x = 10.
```python
age_predict = 10
linear_predict = linear_equation(age_predict)
quadratic_predict = quadratic_equation(age_predict)
exponential_predict = exponential_func(age_predict, a_exp, b_exp)
print(f"Linear Prediction (10 years): ${linear_predict:.2f}")
print(f"Quadratic Prediction (10 years): ${quadratic_predict:.2f}")
print(f"Exponential Prediction (10 years): ${exponential_predict:.2f}")
```
Output:
```
Linear Prediction (10 years): $-2200.00
Quadratic Prediction (10 years): $-8650.00
Exponential Prediction (10 years): $588.64
```
**c) Which Result is Most Reasonable?**
* **Exponential Model:** The exponential model predicts a positive value (around $588.64), which is the most reasonable. Cars don't typically have negative values.
* **Linear and Quadratic Models:** Both the linear and quadratic models predict negative values, which is illogical for a car's value.
**d) Which Function Provides the Best Model?**
* **Exponential Model:** The exponential model is the best fit. Here's why:
* Car values tend to depreciate at a decreasing rate over time, which is characteristic of an exponential decay.
* The exponential model avoids the unrealistic negative values predicted by the other models.
* Exponential models are commonly used to model depreciation.

Question 1171303: NASA is designing a new satellite to go on the international space station. The satellite disk is the shape of a parabola. The satellite will be attached to the station on a pole and will place the vertex of the satellite 4 feet away from the surface of the station. The receiver is to be positioned 7 feet above the roof. Write an equation that best models the parabolic cross section of the satellite dish.( can I see how you solve it please?)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Absolutely! Let's break down how to find the equation for the parabolic cross-section of the satellite dish.
**1. Set up a Coordinate System**
To make this problem easier, let's set up a coordinate system:
* Place the vertex of the parabola at the origin (0, 0).
* Let the axis of symmetry of the parabola be the y-axis.
* Since the receiver is above the vertex, the parabola opens upwards.
**2. Understand the Given Information**
* The vertex is 4 feet away from the surface of the station, and the receiver is 7 feet above the vertex.
* This means the receiver is located at the point (0, 7).
* The standard equation of a parabola that opens upwards with its vertex at the origin is:
* x² = 4py
* Where 'p' is the distance from the vertex to the focus (the location of the receiver).
**3. Find the Value of 'p'**
* In this case, the distance from the vertex (0, 0) to the receiver (focus) (0, 7) is 7 feet.
* Therefore, p = 7.
**4. Write the Equation**
* Substitute p = 7 into the standard equation:
* x² = 4(7)y
* x² = 28y
**Therefore, the equation that best models the parabolic cross-section of the satellite dish is x² = 28y.**
**Visual Explanation**
Imagine the satellite dish in a 2D plane:
* The vertex is at (0, 0).
* The receiver (focus) is at (0, 7).
* The parabola opens upwards along the y-axis.
The equation x² = 28y represents this parabolic shape.

Question 1171448: ``The value of function -10000e^(-0.03x)-120x+10000 at its critical points is
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
To find the value of the function at its critical points, we need to:
1. **Find the derivative of the function.**
2. **Set the derivative equal to zero and solve for x.** This will give us the critical points.
3. **Plug the critical points back into the original function to find the function's value.**
Let's do this step-by-step:
**1. Find the derivative:**
The given function is:
f(x) = -10000e^(-0.03x) - 120x + 10000
The derivative, f'(x), is:
f'(x) = -10000 * (-0.03) * e^(-0.03x) - 120
f'(x) = 300e^(-0.03x) - 120
**2. Set the derivative to zero and solve for x:**
300e^(-0.03x) - 120 = 0
300e^(-0.03x) = 120
e^(-0.03x) = 120 / 300
e^(-0.03x) = 0.4
Take the natural logarithm of both sides:
ln(e^(-0.03x)) = ln(0.4)
-0.03x = ln(0.4)
x = ln(0.4) / -0.03
x ≈ 30.543
**3. Plug the critical point back into the original function:**
f(x) = -10000e^(-0.03x) - 120x + 10000
f(30.543) = -10000e^(-0.03 * 30.543) - 120 * 30.543 + 10000
f(30.543) = -10000e^(-0.91629) - 3665.16 + 10000
f(30.543) = -10000 * 0.4 + 6334.84
f(30.543) = -4000 + 6334.84
f(30.543) = 2334.84
**Therefore, the value of the function at its critical point is approximately 2334.84.**

Question 1171575: A botanist starts with 3 plants. She takes 5 cuttings from each plant
to start new plants. Later, she takes 5 cuttings from each new plant, and so on.
a) Draw a graph of the number of new plants in each of the first 5 rounds of cuttings.
b) Write an equation to model the number of new plants, P, in the nth round of cuttings.
c) How would the graph and equation change in each scenario? Explain.
i) The botanist starts with 10 plants.
ii) In each round, the botanist takes 2 cuttings from each plant.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break this down step by step:
**a) Graph of New Plants**
* **Round 1:**
* Starts with 3 plants.
* Takes 5 cuttings from each: 3 plants * 5 cuttings/plant = 15 new plants.
* **Round 2:**
* Starts with 15 plants.
* Takes 5 cuttings from each: 15 plants * 5 cuttings/plant = 75 new plants.
* **Round 3:**
* Starts with 75 plants.
* Takes 5 cuttings from each: 75 plants * 5 cuttings/plant = 375 new plants.
* **Round 4:**
* Starts with 375 plants.
* Takes 5 cuttings from each: 375 * 5 = 1875
* **Round 5:**
* Starts with 1875 plants.
* Takes 5 cuttings from each: 1875 * 5 = 9375.
Graph:
* X-axis: Round number (1, 2, 3, 4, 5)
* Y-axis: Number of new plants (15, 75, 375, 1875, 9375)
The graph will show an exponential growth curve.
**b) Equation to Model New Plants (P)**
* We see a pattern:
* Round 1: 3 * 5^1 = 15
* Round 2: 3 * 5^2 = 75
* Round 3: 3 * 5^3 = 375
* Round 4: 3*5^4 = 1875
* Round 5: 3*5^5 = 9375
* Therefore, the equation is: P = 3 * 5^n, where P is the number of new plants and n is the round number.
**c) Changes in Graph and Equation**
**i) Starting with 10 Plants:**
* **Equation:**
* The initial factor changes from 3 to 10.
* P = 10 * 5^n
* **Graph:**
* The y-intercept (starting point) of the graph will be higher.
* The exponential growth will be steeper, as there are more plants at each round.
**ii) Taking 2 Cuttings from Each Plant:**
* **Equation:**
* The base of the exponent changes from 5 to 2.
* P = 3 * 2^n
* **Graph:**
* The graph will still show exponential growth, but it will be less steep.
* The number of new plants will increase at a slower rate compared to 5 cuttings per plant.

Question 1172984: A used furniture store buys old mattresses for $50 and sells them for $200. Write an inequality representing all the possible combinations of mattress purchases and sales that will result in a profit of less than $1500. Graph the inequality, and describe an aspect of the used furniture business that an inequality cannot account for.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step.
**1. Define Variables**
* Let 'x' be the number of mattresses purchased.
* Let 'y' be the number of mattresses sold.
**2. Calculate Profit per Mattress**
* Selling price: $200
* Purchase price: $50
* Profit per mattress: $200 - $50 = $150
**3. Write the Profit Inequality**
* Total profit from sales: 150y
* Total cost of purchases: 50x
* Net profit: 150y - 50x
* We want the net profit to be less than $1500:
* 150y - 50x < 1500
**4. Simplify the Inequality**
* Divide the inequality by 50:
* 3y - x < 30
**5. Rewrite in Slope-Intercept Form (for graphing)**
* Add x to both sides:
* 3y < x + 30
* Divide by 3:
* y < (1/3)x + 10
**6. Graph the Inequality**
* **Graph the boundary line:** y = (1/3)x + 10
* Y-intercept: (0, 10)
* Slope: 1/3 (rise 1, run 3)
* **Dashed Line:** Since the inequality is strictly "less than" (<), use a dashed line to indicate that points on the line are *not* included in the solution.
* **Shade the Region:** Shade the region *below* the dashed line, as this represents y < (1/3)x + 10.
* **First Quadrant:** Remember that x and y must be non-negative (you can't buy or sell a negative number of mattresses), so the solution is restricted to the first quadrant.
**7. Aspect the Inequality Cannot Account For**
The inequality only accounts for the financial relationship between purchases, sales, and profit. It *cannot* account for:
* **Demand:** The inequality doesn't consider if there are enough customers willing to buy the mattresses. You might buy many mattresses but not be able to sell them all.
* **Condition of Mattresses:** The inequality assumes all mattresses are sold for $200. Some mattresses might be in poor condition and need repairs, reducing profit, or they might not sell at all.
* **Storage Costs:** The inequality does not factor in the cost of storing the mattresses.
* **Time:** The inequality doesn't consider the time it takes to buy and sell the mattresses. If mattresses sit in storage for a long time, the opportunity cost increases.
* **Market Fluctuations:** The inequality assumes a fixed selling price of $200. Market prices for used mattresses can change.
* **Labor Costs:** The inequality does not factor in the labor costs associated with moving, cleaning, or repairing the mattresses.
* **Disposal Costs:** Some mattresses might be too damaged to sell, and disposing of them could incur a cost.
* **Inventory Limits:** The store may only have so much space for mattresses.

Question 1172982: A used furniture store buys old mattresses for $50 and sell them for $200. Write an inequality representing all the possible combinations of mattress purchases and sales that will result in a profit of less than $1500
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.
A used furniture store buys old mattresses for $50 and sell them for $200.
Write an inequality representing all the possible combinations of mattress
purchases and sales that will result in a profit of less than $1500
~~~~~~~~~~~~~~~~~~~~~~~

Let x be the number of bought mattresses, and let y be the number of sold mattresses. Then the inequality for the profit is -50x + 200y < 1500 dollars. Accompanied inequalities are x >= 9, y >= 0.

Solved.

Question 1174174: ``Your mother wants grapes and bananas for her fruit salad. You found out that grapes sells at Php 200 per kilogram and the banana sells at Php150 per kilogram. How many possible kilogram of each item, you will buy if the total cost is not to exceed Php 500?
Requirement for the Presentation of the Problem.
1. Representation of the given. (Ex. Let x be the no. of apple)
2. Systems of Linear Inequalities
3. Graph of the System of Linear Inequalities
4. Name two solutions of the system.
.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Absolutely! Let's solve this problem step by step.
**1. Representation of the Given**
* Let **x** be the number of kilograms of grapes.
* Let **y** be the number of kilograms of bananas.
* Price of grapes: Php 200 per kilogram
* Price of bananas: Php 150 per kilogram
* Total budget: Php 500
**2. Systems of Linear Inequalities**
* **Cost Inequality:** The total cost of grapes and bananas must not exceed Php 500.
* 200x + 150y ≤ 500
* **Non-negativity Constraints:** You cannot buy negative kilograms of fruit.
* x ≥ 0
* y ≥ 0
**3. Graph of the System of Linear Inequalities**
* **Simplify the Cost Inequality:**
* 200x + 150y ≤ 500
* Divide by 50: 4x + 3y ≤ 10
* **Find the Intercepts of 4x + 3y = 10:**
* If x = 0, then 3y = 10, so y = 10/3 ≈ 3.33
* If y = 0, then 4x = 10, so x = 10/4 = 2.5
* **Graph the Line:** Plot the points (0, 3.33) and (2.5, 0) and draw a line connecting them.
* **Shade the Region:** Since 4x + 3y ≤ 10, shade the region below the line.
* **Graph the Non-negativity Constraints:** Shade the region where x ≥ 0 (right of the y-axis) and y ≥ 0 (above the x-axis).
* **Feasible Region:** The feasible region is the area where all inequalities overlap. This will be a triangle in the first quadrant, bounded by the x-axis, y-axis, and the line 4x + 3y = 10.
**4. Name Two Solutions of the System**
* **Solution 1:**
* x = 1 kg of grapes
* y = 2 kg of bananas
* Cost: (200 * 1) + (150 * 2) = 200 + 300 = Php 500 (within the budget)
* This point (1,2) is within the feasible region.
* **Solution 2:**
* x = 0.5 kg of grapes
* y = 2 kg of bananas
* Cost: (200 * 0.5) + (150 * 2) = 100 + 300 = Php 400 (within the budget)
* This point (0.5,2) is within the feasible region.

Question 1174850: An airline company has drawn up a new flight schedule involving five flights. To assist
in allocating five pilots to the flights, it has asked them to state their preference scores by giving each flight a number out of 10 .The higher the number , the greater is the preference.
Certain of these flights are unsuited to some pilots owing to domestic reasons. These have
been marked with a X
a b C d e
A 8 2 x 5 4
B 10 9 2 8 4
C 5 4 9 6 x
D 3 6 2 8 7
E 5 6 10 4 3
Flight number
What should be the allocation of the pilots to flights in order to meet as many performances as
possible? (Hint: The problem is to maximize the total preference score).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
This is a classic assignment problem that can be solved using the Hungarian Algorithm. Since we need to maximize the total preference score, we'll need to negate the preference scores to convert it to a minimization problem.
**1. Prepare the Cost Matrix**
First, replace the 'x' entries with a very low number (e.g., -100) to ensure they are not selected. Then, negate all the preference scores to convert the maximization problem to a minimization problem.
```
a b c d e
A -8 -2 100 -5 -4
B -10 -9 -2 -8 -4
C -5 -4 -9 -6 100
D -3 -6 -2 -8 -7
E -5 -6 -10 -4 -3
```
**2. Apply the Hungarian Algorithm**
* **Row Reduction:** Subtract the smallest element in each row from all elements in that row.
```
a b c d e
A -6 6 108 -3 -2
B -8 -7 0 -6 -2
C -1 0 -5 -2 104
D 1 -3 0 -6 -5
E -2 -3 -7 -1 0
```
* **Column Reduction:** Subtract the smallest element in each column from all elements in that column.
```
a b c d e
A -5 9 108 -2 -2
B -7 -4 0 -5 -2
C 0 3 -5 -1 104
D 2 0 0 -5 -5
E 0 0 -7 0 0
```
* **Cover Zeros:** Draw the minimum number of horizontal and vertical lines to cover all zeros.
```
a b c d e
A -5 9 108 -2 -2
B -7 -4 0 -5 -2
C 0 3 -5 -1 104
D 2 0 0 -5 -5
E 0 0 -7 0 0
| | --- | ---
```
* **Optimal Allocation:** Since we have five lines covering the zeros, we can make an optimal allocation.
* **Find Optimal Assignments:**
* Start by looking for rows or columns with single zeros.
* Pilot E: E-e
* Pilot D: D-c
* Pilot C: C-a
* Pilot B: B-b
* Pilot A: A-d
**3. Convert Back to Preference Scores**
* A-d: 5
* B-b: 9
* C-a: 5
* D-c: 2
* E-e: 3
**4. Calculate Total Preference Score**
* Total = 5 + 9 + 5 + 2 + 3 = 24
**Allocation**
* Pilot A: Flight d
* Pilot B: Flight b
* Pilot C: Flight a
* Pilot D: Flight c
* Pilot E: Flight e
**Total Preference Score: 24**

Question 1175464: Consider the following grouped frequency distribution, which portrays the duration (in minutes) of 50 lectures
delivered by a Chemistry professor.
Duration (Minutes) Frequency
30.00 < 40.00 3
40.00 < 50.00 7
50.00 < 60.00 18
60.00 < 70.00 13
70.00 < 80.00 8
80.00 < 90.00 1

Using computational formulae, calculate the:
(i) estimated mean.
(ii) mode.
(iii) standard deviation.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's calculate the estimated mean, mode, and standard deviation for this grouped frequency distribution.
**1. Estimated Mean**
* **Midpoints (x_m):** Find the midpoint of each class interval.
* 30-40: 35
* 40-50: 45
* 50-60: 55
* 60-70: 65
* 70-80: 75
* 80-90: 85
* **Multiply Midpoints by Frequencies (f * x_m):**
* 35 * 3 = 105
* 45 * 7 = 315
* 55 * 18 = 990
* 65 * 13 = 845
* 75 * 8 = 600
* 85 * 1 = 85
* **Sum of (f * x_m):** 105 + 315 + 990 + 845 + 600 + 85 = 2940
* **Sum of Frequencies (n):** 3 + 7 + 18 + 13 + 8 + 1 = 50
* **Estimated Mean (x̄):** (Σ f * x_m) / n = 2940 / 50 = 58.8
**(i) Estimated Mean = 58.8 minutes**
**2. Mode**
* **Modal Class:** The class with the highest frequency is 50-60 (frequency = 18).
* **Lower Boundary (L):** 50
* **Class Width (w):** 10
* **Frequency of Modal Class (f_m):** 18
* **Frequency of Class Before Modal Class (f_1):** 7
* **Frequency of Class After Modal Class (f_2):** 13
* **Mode Formula:** L + [(f_m - f_1) / ((f_m - f_1) + (f_m - f_2))] * w
* Mode = 50 + [(18 - 7) / ((18 - 7) + (18 - 13))] * 10
* Mode = 50 + [11 / (11 + 5)] * 10
* Mode = 50 + (11 / 16) * 10
* Mode = 50 + (0.6875) * 10
* Mode = 50 + 6.875
* Mode = 56.875
**(ii) Mode = 56.875 minutes**
**3. Standard Deviation**
* **Calculate (x_m - x̄)²:**
* (35 - 58.8)² = (-23.8)² = 566.44
* (45 - 58.8)² = (-13.8)² = 190.44
* (55 - 58.8)² = (-3.8)² = 14.44
* (65 - 58.8)² = (6.2)² = 38.44
* (75 - 58.8)² = (16.2)² = 262.44
* (85 - 58.8)² = (26.2)² = 686.44
* **Multiply by Frequencies (f * (x_m - x̄)²):**
* 3 * 566.44 = 1699.32
* 7 * 190.44 = 1333.08
* 18 * 14.44 = 259.92
* 13 * 38.44 = 499.72
* 8 * 262.44 = 2099.52
* 1 * 686.44 = 686.44
* **Sum of (f * (x_m - x̄)²):** 1699.32 + 1333.08 + 259.92 + 499.72 + 2099.52 + 686.44 = 6578.00
* **Variance (s²):** (Σ f * (x_m - x̄)²) / (n - 1) = 6578.00 / (50 - 1) = 6578 / 49 = 134.2449
* **Standard Deviation (s):** √Variance = √134.2449 ≈ 11.5864
**(iii) Standard Deviation ≈ 11.59 minutes**

Question 1176522: 1. You sell peanuts at UNK athletic events to make some extra money. When peanuts are sold for $1.00 per bag, approximately 600 bags are sold at each event. You tried to raise the price to $1.25 and found that the quantity demanded dropped to 560 bags. The startup cost was $500 and the cost to you per bag of peanuts is $0.50.
A. Assume that the demand function is linear and write a function that models the profit from selling x bags of peanuts at an event.
B. What number of bags and what price per bag will get you a maximum profit?
C. Does your answer to part B agree with what you "thought" it would be? Why or why not?
Even telling me what kind of question this is so I can find other examples would be helpful, thanks!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's solve this problem step-by-step.
**A. Model the Profit Function**
1. **Find the Demand Function (p(x))**
* We have two points: (600, 1.00) and (560, 1.25)
* Find the slope (m):
* m = (1.25 - 1.00) / (560 - 600) = 0.25 / -40 = -0.00625
* Use point-slope form (y - y1 = m(x - x1)):
* p - 1.00 = -0.00625(x - 600)
* p = -0.00625x + 3.75 + 1.00
* p(x) = -0.00625x + 4.75
2. **Find the Revenue Function (R(x))**
* Revenue = price * quantity
* R(x) = x * p(x)
* R(x) = x(-0.00625x + 4.75)
* R(x) = -0.00625x² + 4.75x
3. **Find the Cost Function (C(x))**
* Cost = startup cost + cost per bag * quantity
* C(x) = 500 + 0.50x
4. **Find the Profit Function (P(x))**
* Profit = Revenue - Cost
* P(x) = R(x) - C(x)
* P(x) = (-0.00625x² + 4.75x) - (500 + 0.50x)
* P(x) = -0.00625x² + 4.25x - 500
**B. Maximize Profit**
1. **Find the Vertex of the Profit Function**
* The profit function is a quadratic, so its maximum occurs at the vertex.
* The x-coordinate of the vertex is given by x = -b / 2a, where a = -0.00625 and b = 4.25.
* x = -4.25 / (2 * -0.00625)
* x = -4.25 / -0.0125
* x = 340
2. **Find the Price per Bag**
* p(x) = -0.00625x + 4.75
* p(340) = -0.00625(340) + 4.75
* p(340) = -2.125 + 4.75
* p(340) = 2.625
3. **Find the Maximum Profit**
* P(x) = -0.00625x² + 4.25x - 500
* P(340) = -0.00625(340)² + 4.25(340) - 500
* P(340) = -0.00625(115600) + 1445 - 500
* P(340) = -722.5 + 1445 - 500
* P(340) = 222.5
**Answers for B:**
* Number of bags: 340
* Price per bag: $2.625 (or $2.63)
**C. Agreement with Intuition**
* **Initial Intuition:** You might have thought that raising the price would always decrease the number of bags sold and potentially lower the profit.
* **Actual Result:** The analysis shows that there's an optimal price point that maximizes profit. In this case, raising the price significantly above the initial $1.00 leads to a lower quantity sold but a higher profit.
* **Explanation:** This is because the higher price per bag more than compensates for the reduced quantity sold, up to a certain point. The profit function is a parabola, and the vertex represents the optimal balance between price and quantity.
**Why the "thought" might not be what the answer is:**
* **Linear Demand Assumption:** The linear demand function is a simplification. Real-world demand might not be perfectly linear.
* **Cost Structure:** The constant cost per bag and fixed startup cost simplify the model. Real costs might be more complex.
* **Consumer Behavior:** The model assumes rational consumer behavior. In reality, factors like brand loyalty, perceived value, and competitor pricing can influence demand.
In conclusion, the mathematical analysis provides a more precise and optimal solution than relying on initial intuition alone.

Question 1177453: In order to raise money for team towels, your swim team decides to sell pizza and subs. The profits are $2.75 and $1.45 for for each sub. You sell 48 subs. Write and solve an inequality to determine the number of pizzas you must sell to have a total profit of at least $100.
Your calling card charges $1.25 to place a call plus $0.35 for each minute. Write and solve an inequality to determine how long a call can last so that the total cost stays below $5.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**Pizza and Subs**
Let's break down the pizza and subs problem:
**1. Define a variable:**
* Let 'p' be the number of pizzas you need to sell.
**2. Write the inequality:**
* Profit from pizzas: $2.75p
* Profit from subs: $1.45 * 48 = $69.60
* Total profit: $2.75p + $69.60
* We want the total profit to be at least $100, so the inequality is:
$2.75p + $69.60 ≥ $100
**3. Solve the inequality:**
* Subtract $69.60 from both sides: $2.75p ≥ $30.40
* Divide both sides by $2.75: p ≥ 11.05
Since you can't sell parts of pizzas, you need to sell at least 12 pizzas.
**Answer:** You must sell at least 12 pizzas to have a total profit of at least $100.

**Calling Card**
Now let's solve the calling card problem:
**1. Define a variable:**
* Let 'm' be the number of minutes the call can last.
**2. Write the inequality:**
* Cost of the call: $1.25 + $0.35m
* We want the total cost to stay below $5, so the inequality is:
$1.25 + $0.35m < $5
**3. Solve the inequality:**
* Subtract $1.25 from both sides: $0.35m < $3.75
* Divide both sides by $0.35: m < 10.71
**Answer:** The call can last for a maximum of 10 minutes to keep the total cost below $5.

Question 1209774: Two numba ?*?=-5 and when added +5
Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.

Starting equations x *y = -5 (1) x + y = 5 (2) From equation (2), express y = 5 - x and substitute it into equation (1). x*(5-x) = -5. Simplify 5x - x^2 = -5, x^2 - 5x - 5 = 0. Solve quadratic equation using the quadratic formula = = = . The numbers are = 5.854101966 and -0.854101966. ANSWER CHECK. 5.854101966*(-0.854101966) = -5.0000000000 with at least 10 decimals after the decimal dot.

Solved.

Question 1178966: 1. Owen would like to make a small income as an artist. Owen asked his friend Emily for advice about what
combination of pictures to make. She suggested that he determine a reasonable profit for that month’s work
and then paint what he needs in order to earn that amount of profit.
• Each pastel requires $5 in materials and earns a profit of $40 for Owen.
• Each watercolor requires $15 in materials and earns a profit of $105 for Owen.
• Owen has $180 to spend on materials.
• Owen can make at most 16 pictures.
a. State the system of inequalities that represents this situation. Remember to define your variables
and include any non-negative constraints that are required. (4 marks)
b. What is the optimization equation? (1 mark)
c. On graph paper, create this feasible region to use in this problem. Label your axes. (4 marks)
d. Suppose Owen decided $1,000 would be a reasonable profit. Find three different combinations of
watercolors and pastels that would earn Owen a profit of exactly $1,000. (3 marks)
e. Now suppose Owen wanted to earn only $500 in profit. Find three different combinations of
watercolors and pastels that will earn Owen a profit of exactly $500. Using a different-coloured
pencil, add those points to your graph. (3 marks)
f. Owen’s mother has convinced him that he should try to earn as much as possible. So, Owen needs
to figure out the most profit he can earn within his constraints. He also wants to be able to prove to
his mother that it is really the maximum amount. Find the maximum possible profit that Owen can
earn and the combination of pictures he should make to earn that profit.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step:
**a. State the System of Inequalities:**
* **Variables:**
* Let 'p' represent the number of pastels.
* Let 'w' represent the number of watercolors.
* **Material Cost Constraint:**
* 5p + 15w ≤ 180 (Owen has $180 to spend on materials)
* **Total Pictures Constraint:**
* p + w ≤ 16 (Owen can make at most 16 pictures)
* **Non-Negative Constraints:**
* p ≥ 0 (Owen cannot make a negative number of pastels)
* w ≥ 0 (Owen cannot make a negative number of watercolors)
**b. What is the Optimization Equation?**
* **Profit Equation:**
* Profit = 40p + 105w (Owen earns $40 per pastel and $105 per watercolor)
**c. Create the Feasible Region (Graph):**
1. **Graph 5p + 15w ≤ 180:**
* Rewrite as: p + 3w ≤ 36
* Find intercepts:
* If p = 0, 3w = 36, w = 12 (point: (0, 12))
* If w = 0, p = 36 (point: (36, 0))
* Draw a line through these points. Shade below the line.
2. **Graph p + w ≤ 16:**
* Find intercepts:
* If p = 0, w = 16 (point: (0, 16))
* If w = 0, p = 16 (point: (16, 0))
* Draw a line through these points. Shade below the line.
3. **Graph p ≥ 0 and w ≥ 0:**
* This restricts the feasible region to the first quadrant.
4. **Feasible Region:** The area where all shaded regions overlap.
* Find the corner points where the lines intersect.
* (0, 0)
* (16, 0)
* (0, 12)
* Intersection of p + 3w = 36 and p + w = 16:
* Subtract the equations: 2w = 20, w = 10
* Substitute w = 10 into p + w = 16: p = 6 (point: (6, 10))
**d. Combinations for $1,000 Profit:**
* 1000 = 40p + 105w
* We need to find integer solutions for p and w.
* 1000/5 = 200, so 40p + 105w must be divisible by 5. 105w is always divisible by 5, so 40p must be divisible by 5.
* **Combination 1:** If w = 0, 40p = 1000, p = 25. (25, 0)
* **Combination 2:** If w = 4, 40p + 420 = 1000, 40p = 580, p = 14.5. (not integer)
* **Combination 3:** If w = 8, 40p + 840 = 1000, 40p = 160, p = 4. (4, 8)
* **Combination 4:** if w=12, 105w=1260. too high.
* **Combination 5:** if p = 10, 400 + 105w = 1000, 105w = 600. w= 5.7. not integer.
* **Combination 6:** if p=20, 800+105w=1000, 105w=200, not integer.
* Therefore: (25, 0), (4, 8), (10, 5.7) is not a solution.
* (25, 0)
* (4, 8)
* (10, 5.7) is not a solution, but (10, 6) will be slightly over 1000.
**e. Combinations for $500 Profit:**
* 500 = 40p + 105w
* We need to find integer solutions for p and w.
* **Combination 1:** If w = 0, 40p = 500, p = 12.5. (not integer)
* **Combination 2:** If w = 2, 40p + 210 = 500, 40p = 290, p = 7.25. (not integer)
* **Combination 3:** If w = 4, 40p + 420 = 500, 40p = 80, p = 2. (2, 4)
* **Combination 4:** If p = 5, 200 + 105w = 500, 105w = 300, w = 2.85. not integer.
* **Combination 5:** If p = 10, 400 + 105w = 500, 105w = 100, not integer.
* Therefore: (2, 4), (12.5, 0), (5, 2.85) is not a solution.
* (2, 4)
* (12.5,0)
* (5, 2.85) is not a solution.
**f. Maximum Profit:**
* Evaluate the profit equation at the corner points of the feasible region:
* (0, 0): Profit = 40(0) + 105(0) = 0
* (16, 0): Profit = 40(16) + 105(0) = 640
* (0, 12): Profit = 40(0) + 105(12) = 1260
* (6, 10): Profit = 40(6) + 105(10) = 240 + 1050 = 1290
* **Maximum Profit:** $1290
* **Combination:** 6 pastels and 10 watercolors.

Question 1181397: Professor Weejohs teaches two sections of business analytics, which combined will result in 120 final exams to be graded. Professor Weejohs has two assistants, Ralph and Melanie, who will grade the final exams. There is a 3-day period between the time the exam is administered and when final grades must be posted. During this period Ralph has 12 hours available and Melanie has 10 hours available to grade the exams. It takes Ralph an average of 7.2 minutes to grade an exam, and it takes Melanie 12 minutes to grade an exam; however, Ralph’s exams will have errors that will require Professor Weejohs to ultimately regrade 10% of the exams, while only 6% of Melanie’s exams will require regrading. Professor Weejohs wants to know how many exams to assign to each assistant to grade in order to minimize the number of exams to regrade.
Answer by mccravyedwin(406)   (Show Source):
You can put this solution on YOUR website!

Let x be the number of exams to assign Melanie. Then since there are 120 exams, Ralph will be assigned 120-x. Inequality for Melanie's grading time: 12x minutes = 0.2x hours, which must be less than or equal to 10 hours, or 0.2x <= 10 x <= 50 Inequality for Ralph's grading time: 7.2(120-x) minutes = 0.12(120-x) hours, which must be less than or equal to 12 hours, or 0.12(120-x) <= 12 120-x <= 100 -x <= -20 x >= 20 So 20 <= x <= 50 We want to minimize C = 0.06x + 0.10(120-x) = 0.06x + 12 - 0.10x = 12.06 - 0.04x, which will be when we choose x = 50 (in order to subtract the most from 12.06.) So the Professor will assign 50 exams to Melanie and the other 120-50=70 to Ralph so that the professor will only need to re-grade 10.06 (about 10) exams. Answer: The professor should assign 50 exams to Melanie and the other 70 to Ralph. Edwin

Question 1184064: During one month, a homeowner used 700 units of electricity and 100 units of gas for a total cost of $486. The next month, 500 units of electricity and 250 units of gas were used for a total cost of $415. Find the cost per unit of gas.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem:
**Let:**
* 'e' be the cost per unit of electricity.
* 'g' be the cost per unit of gas.
**Set up the equations:**
* **Equation 1 (Month 1):** 700e + 100g = 486
* **Equation 2 (Month 2):** 500e + 250g = 415
**Solve the equations:**
One way to solve this is using elimination:
1. **Multiply Equation 1 by 2.5:**
1750e + 250g = 1215
2. **Subtract Equation 2 from the modified Equation 1:**
(1750e + 250g) - (500e + 250g) = 1215 - 415
1250e = 800
3. **Solve for e:**
e = 800 / 1250
e = 0.64
4. **Substitute the value of e back into either Equation 1 or 2 to solve for g. Let's use Equation 1:**
700(0.64) + 100g = 486
448 + 100g = 486
100g = 38
g = 0.38
**Answer:**
The cost per unit of gas is $0.38.

Question 1184067: A state employees' pension fund invested a total of one million dollars in two accounts that earned 3.5% and 4.5% annual simple interest. At the end of the year, the total interest earned from the two investments was $43,000. How much was invested at each rate?

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem:
**Let:**
* 'x' be the amount invested at 3.5% interest.
* 'y' be the amount invested at 4.5% interest.
**Set up the equations:**
* **Equation 1 (Total investment):** x + y = 1,000,000
* **Equation 2 (Total interest):** 0.035x + 0.045y = 43,000
**Solve the equations:**
One way to solve this is using substitution:
1. **Solve Equation 1 for x:** x = 1,000,000 - y
2. **Substitute this value of x into Equation 2:**
0.035(1,000,000 - y) + 0.045y = 43,000
3. **Simplify and solve for y:**
35,000 - 0.035y + 0.045y = 43,000
0.01y = 8,000
y = 800,000
4. **Substitute the value of y back into Equation 1 to find x:**
x + 800,000 = 1,000,000
x = 200,000
**Answer:**
* $200,000 was invested at 3.5% interest.
* $800,000 was invested at 4.5% interest.

Question 1184068: In investing $5,650 of a couple's money, a financial planner put some of it into a savings account paying 4% annual simple interest. The rest was invested in a riskier mini-mall development plan paying 10% annual simple interest. The combined interest earned for the first year was $445. How much money was invested at each rate?

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem:
**Let:**
* 'x' be the amount invested in the savings account at 4% interest.
* 'y' be the amount invested in the mini-mall development at 10% interest.
**Set up the equations:**
* **Equation 1 (Total investment):** x + y = 5650
* **Equation 2 (Total interest):** 0.04x + 0.10y = 445
**Solve the equations:**
One way to solve this is using substitution:
1. **Solve Equation 1 for x:** x = 5650 - y
2. **Substitute this value of x into Equation 2:**
0.04(5650 - y) + 0.10y = 445
3. **Simplify and solve for y:**
226 - 0.04y + 0.10y = 445
0.06y = 219
y = 3650
4. **Substitute the value of y back into Equation 1 to find x:**
x + 3650 = 5650
x = 2000
**Answer:**
* $2000 was invested in the savings account at 4% interest.
* $3650 was invested in the mini-mall development at 10% interest.

Question 1185383: The design costs of an advertisement in a glossy magazine are £9000 and the cost per cm2 of print is
£50.
a. Write an expression for the total cost of publishing an advert which covers x cm2
b. The advertising budget is between £10,800 and £12,500. Write down and solve an inequality to
work out the minimum and maximum area that could be used.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**a.** Let 'C' be the total cost and 'x' be the area in cm². The expression for the total cost is:
C = 9000 + 50x
**b.** The advertising budget is between £10,800 and £12,500. So, the inequality is:
10800 ≤ C ≤ 12500
Substitute the expression for C:
10800 ≤ 9000 + 50x ≤ 12500
Subtract 9000 from all parts of the inequality:
10800 - 9000 ≤ 50x ≤ 12500 - 9000
1800 ≤ 50x ≤ 3500
Divide all parts by 50:
1800/50 ≤ x ≤ 3500/50
36 ≤ x ≤ 70
Therefore, the minimum area that could be used is 36 cm², and the maximum area is 70 cm².

Question 1184066: A company manufactures both mountain bikes and trail bikes. The cost of materials for a mountain bike is $60, and the cost of materials for a trail bike is $40. The cost of labor to manufacture a mountain bike is $90, and the cost of labor to manufacture a trail bike is $30. During a week in which the company has budgeted $1,700 for materials and $1,950 for labor, how many mountain bikes does the company plan to manufacture?

Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.
A company manufactures both mountain bikes and trail bikes.
The cost of materials for a mountain bike is $60, and the cost of materials for a trail bike is $40.
The cost of labor to manufacture a mountain bike is $90, and the cost of labor to manufacture a trail bike is $30.
During a week in which the company has budgeted $1,700 for materials and $1,950 for labor,
how many mountain bikes does the company plan to manufacture?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let m be the number of the mountain bikes, t be the number of the trail bikes. Write equations as you read the problem 60m + 40t = 1700 (1) (material money) 90m + 30t = 1950 (2) (labor money) Solve by the elimination method. Since they ask about mountain bikes, eliminate t. For it, multiply equation (1) by 3 and multiply equation (2) by 4. You will get 180m + 120t = 5100 (3) 360m + 120t = 7800 (4) Now subtract equation (3) from equation(4) 180m = 7800 - 5100 180m = 2700 m = 2700/180 = 15. ANSWER. The company plans to manufacture 15 mountain bikes.

Solved.

Question 1209578: the sum of two numbers is 92. their difference is 20. find the numbers
Found 3 solutions by greenestamps, timofer, ikleyn:
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!

Here is another informal solution method that can be done mentally.

Consider the numbers on a number line.

You start at the first number and move one direction a distance equal to the second number and end up at 92; or you move the other direction the same distance and end up at 20.

That means the first number is halfway between 92 and 20 -- i.e., it is the average of 92 and 20. So the first number is (92+20)/2 = 56.

Then the second number is the number you need to add to 56 to reach 92: 92-56 = 36.

ANSWERS: 56 and 36

Answer by timofer(104)   (Show Source):
You can put this solution on YOUR website!

same as

36 and 56

Answer by ikleyn(52777)   (Show Source):
You can put this solution on YOUR website!
.

Mentally in your mind, make your greater number 20 units less.

You will have two equal numbers with the sum of 92 - 20 = 72.

Hence, the smaller of the two original numbers is 72/2 = 36.

Then the greater of the two original numbers is 36 + 20 = 56.

ANSWER.   The numbers are 36 and 56.

To complete the solution in full, check mentally that the difference of the found numbers is 20 and the sum is 92.

Solved mentally.

--------------------

I wrote this post to show you that this simple problem
can be solved mentally without using equations.

As well as a million of other similar simple problems.

Question 1186442: Assume the number of U.S. dial-up Internet households stood at 42.5 million at the beginning of 2004 and was projected to decline at the rate of 3.9 million households per year for the next 7 years.
(a) Find a linear function f giving the projected U.S. dial-up Internet households (in millions) in year t, where t = 0 corresponds to the beginning of 2004.
(b) What is the projected number of U.S. dial-up Internet households at the beginning of 2011?

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**(a) Finding the linear function:**
Since the number of households is declining at a constant rate, we can model this situation with a linear function of the form:
f(t) = mt + b
where:
* f(t) is the projected number of households (in millions) in year t
* m is the slope (rate of change)
* b is the y-intercept (initial number of households)
We are given:
* b = 42.5 million (initial number of households at t = 0)
* m = -3.9 million households per year (decline rate, so it's negative)
Therefore, the linear function is:
f(t) = -3.9t + 42.5
**(b) Projecting households at the beginning of 2011:**
Since t = 0 corresponds to the beginning of 2004, the beginning of 2011 corresponds to t = 2011 - 2004 = 7.
We can now plug t = 7 into our linear function:
f(7) = -3.9 * 7 + 42.5
f(7) = -27.3 + 42.5
f(7) = 15.2
So, the projected number of U.S. dial-up Internet households at the beginning of 2011 is 15.2 million.

Question 1209542: For homework, Randall is solving the system of linear equations in the box. 3x + 4y = 25 4x - 7y = - 16 Which strategy should Randall use to solve this system of equations?
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website!
Which strategy depends on what he knows. Upon appearance, substitution seems not the best choice but it can work.

Elimination Method:

ADD:

But recheck this part in case of mistakes in the work.

Question 1186916: Supposed that an earthquake released 10^6 joules of energy.What is the magnitude on a richr scale?How much more energy released that the reference earthquake?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**1. Calculate the Richter scale magnitude:**
The formula relating energy released (E) to Richter scale magnitude (M) is:
log₁₀(E) = 4.4 + 1.5M
Where:
* E is the energy released in joules
* M is the Richter scale magnitude
Plug in the given energy (E = 10⁶ joules) and solve for M:
log₁₀(10⁶) = 4.4 + 1.5M
6 = 4.4 + 1.5M
1.6 = 1.5M
M = 1.6 / 1.5
M ≈ 1.07
Therefore, an earthquake releasing 10⁶ joules of energy has a magnitude of approximately **1.07** on the Richter scale.
**2. Calculate the energy released relative to the reference earthquake:**
The Richter scale is a logarithmic scale. Each whole number increase in magnitude represents a tenfold increase in amplitude and approximately 31.6 times more energy released.
The reference earthquake for the Richter scale is assigned a magnitude of 0. Let E₀ be the energy released by the reference earthquake.
Using the formula:
log₁₀(E₀) = 4.4 + 1.5 * 0
log₁₀(E₀) = 4.4
E₀ = 10⁴.⁴ joules
To find how much more energy the given earthquake released compared to the reference earthquake, we can divide the energy released by the given earthquake by the energy released by the reference earthquake:
Energy ratio = E / E₀ = 10⁶ / 10⁴.⁴ = 10¹·⁶ ≈ 39.81
Therefore, the earthquake released approximately **39.81 times** more energy than the reference earthquake.