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Question 1158623: Determine the location and value of the absolute extreme values of f on the given interval, if they exist.
f(x) =4sqrt(x) −8x^2/x
on [1,10]
Answer by KMST(5396)

(Show Source):

You can put this solution on YOUR website!
The function f(x)=4sqrt(x)−8x^2/x is not a very interesting one, and may not the the function intended.
I suspect the function intended could have been f(x)=(4sqrt(x)-8x^2)/x , which means f%28x%29

, also not an interesting function.
The domain of f(x)=(4sqrt(x)-8x^2)/x is x%3E0

, which can be expressed as %22%280%2C%22

.
The graph of f%28x%29

%22=%22%7D%7D%7B%7B%7B%284sqrt%28x%29-8x%5E2%29%2Fx

graph%28300%2C200%2C-2%2C13%2C-90%2C10%2C%284sqrt%28x%29-8x%5E2%29%2Fx%29

The derivative is negative throughout the domain of the function, meaning that the function decreases continuously.
Its absolute extremes in the interval [1, 10] are
f%281%29

, a maximum, and
f%2810%29

, a minimum, with a rounded value of highlight%28-67.35%29

The domain of f(x)=4sqrt(x)−8x^2/x is %22%280%2C%22

.
That function is f%28x%29=4sqrt%28x%29

Its graph is graph%28300%2C200%2C-2%2C13%2C-90%2C10%2C4sqrt%28x%29-8x%29

The derivative is negative throughout the domain of the function, meaning that the function decreases continuously.
Its absolute extremes in the interval [1, 10] are
f%281%29=4sqrt%281%29-8%2A1=4%2A1-8=4-8=highlight%28-4%29

f%281%29=4sqrt%281%29-8%2A1=4%2A1-8=4-8=highlight%28-4%29

, a maximum, and
f%2810%29=4sqrt%2810%29-8%2A10=highlight%284sqrt%2810%29-8%29

, a minimum, with a rounded value of highlight%28-78.735%29

Question 1210637: add a solver primarily targeted for Pythagorean theorem?
Answer by ikleyn(53937)

(Show Source):

You can put this solution on YOUR website!
.
add a solver primarily targeted for Pythagorean theorem?
~~~~~~~~~~~~~~~~~~~~~~~~~~~

This forum just HAS several such solvers.

See the section Geometry, topic "Pythagorean theorem",
folder "Solvers".

They were created 20 years ago.

Happy learning !

Question 956874: if $8000 is invested in a 2.5% account for 10 years compounded continuously how long would it take for the account to double?
Answer by ikleyn(53937)

(Show Source):

You can put this solution on YOUR website!
.
if $8000 is invested in a 2.5% account for 10 years compounded continuously
how long would it take for the account to double?
~~~~~~~~~~~~~~~~~~~~~~~~~~~

In his post, @lwsshak3 gives the answer "28 years".

His calculation formula t = ln%282%29%2F0.025

is correct, and the value which this formula produces is 27.72588722.

So the expected answer is 27.726 years, or 28 years and 265 days, approximately.

It is the expected answer <<<--->>> not 28 years, as it is given in the post by @lwsshar3.
In this problem, we should not round to the closest greater year, because compounding is continuous.

In such problems, it is important to give adequate answers in accordance with the common sense
to demonstrate to the teacher and to all around that you correctly understand the problem.

(In simple terms - to demonstrate that you are not an idiot).

Question 968430: What is the exact value of arccos(cos(3π/2) )
Found 2 solutions by n2, ikleyn:
Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
What is the exact value of arccos(cos(3π/2) )
~~~~~~~~~~~~~~~~~~~~~~~

This problem is a simple elementary . It provokes you to answer " ", but this answer is WRONG.

To answer correctly, you should remember that function arccos has values in interval [,]. Therefore, you should transform to the equivalent angle in interval [,]. This equivalent angle is , so the correct answer to the problem's question is .

Solved.

Be accurate and attentive in order for do not fall in this and similar traps.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
What is the exact value of arccos(cos(3π/2) )
~~~~~~~~~~~~~~~~~~~~~~~

        The answer "3pi/2" in the post by @lwsshak3 is INCORRECT.

This problem is a simple elementary , and @lwsshak3 fell in this trap. To answer correctly, you should remember that function arccos has values in interval [,]. Therefore, you should transform to the equivalent angle in interval [,]. This equivalent angle is , so the correct answer to the problem's question is .

        Solved correctly, while the "solution" by @lwsshak3 is disproved.

Question 968423: Find the value of tan x given that sin x= 3/5 and x is in the 1st quadrant.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Find the value of tan x given that sin x= 3/5 and x is in the 1st quadrant.
~~~~~~~~~~~~~~~~~~~~~~~~~~

The answer in the post by @lwsshak3 "tan(x) = 3/5" is INCORRECT.

The correct answer is tan(x) = 3/4.

Question 1157973: When ax³ + bx² + cx - 4 is divided by (x+2), the remainder is double that obtained when the expression is divided by (x+1). Show that c can have any value and find b in terms of a.
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!

If you divide by , you obtain a quotient and remainder that is a constant.
You may remember that in then
If you did not remember, you would understand that if the reminder is ,
it means that and that
for , .
So --> .
Similarly the remainder, when dividing by is
--> .
Then, -->-->-->-->

Question 971377: Find all solutions of each equation on the interval(0, 2pi)
Sec^2 x + 2tan x = 0
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Find all solutions of each equation on the interval (0, 2pi)
Sec^2 x + 2tan x = 0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @lwsshak3 is incorrect (incomplete).
        He provided only one root of the equation in his answer, and gave incorrect value for it.
        while the problem has two roots. So, one root was determined incorrectly, while the second root was missed.

        I came to bring a correct solution.

sec^2 x + 2tan x = 0 lcd: cos^2(x) 1 + 2sin(x)*cos(x) = 0 1 + sin(2x) = 0 sin(2x) = -1 2x = , k = 0, +/-1, +/-2, . . . We take only two values 2x = (k=0) and 2x = (k=1), since other values produce values of x out of the given interval. ANSWER. The given equation has two roots in the interval [,). They are , or 135 degrees, and , or 315 degrees.

Solved correctly.

Question 1162355: A large bakery regularly orders cartons of blueberries. Random sample of 50 cartons from each
supplier were weighted. The weights in grams of the cartons were shown in following frequency
distribution.
-------------Supplier A ---------------------Supplier B-----------
Weight of cartons ---Frequency ----Weight of cartons ----Frequency
...501 - 550 ............5 .................501 - 550 .............2
...551 - 600 ...........10................. 551 - 600 .............8
...601 - 650 ...........20 .................601 - 650 ............12
...651 - 700 ...........12 .................651 - 700 ............20
...701 - 750 ............3 .................701 - 750 .............8
a) State the variable of interest. Indicate whether the variable is quantitative discrete,
quantitative continuous or qualitative. (3 marks)
b) Construct suitable graph to represent the above frequency distribution of the weight of
cartons for supplier A and supplier B, respectively. The weight of cartons from which
supplier weight more? Justify. (12 marks)
c) Refer to the graph in (b), suggest the best measure of central tendency and a measure of
dispersion that is suitable to represent the set of data. Justify your answers. (4 marks)
d) Compute all the measures of central tendency and standard deviation for the weight of
cartons from supplier A and interpret the values. (18 marks)
e) Draw ogive and compute the quartile deviation for weight of cartons from supplier B.
(10 marks)
f) The bakery uses one carton of blueberries in each blueberry muffin recipe. It is important
that the cartons be of consistent weight so that the muffins turn out right. Which supplier
should the bakery choose? Explain why. (3 marks)
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
This analysis compares the blueberry carton weights from two suppliers to determine which is better suited for a bakery's needs.
### a) Variable Analysis
* **Variable of Interest:** The **weight** of the blueberry cartons (in grams).
* **Classification:** This is a **quantitative continuous** variable (weight can be measured to infinitely precise decimal values, even though it is grouped here).
---
### b) Graphical Representation and Comparison
To represent this data, a **Histogram** or **Frequency Polygon** is most suitable. By comparing the distributions:
* **Supplier A:** The peak (mode) is in the $601 - 650$g range ($20$ cartons).
* **Supplier B:** The peak (mode) is in the $651 - 700$g range ($20$ cartons).
**Justification:** **Supplier B** weighs more on average. Visually, its distribution is shifted to the right compared to Supplier A. The majority of Supplier B’s cartons ($28$ out of $50$) weigh over $650$g, whereas only $15$ of Supplier A’s cartons fall into those heavier categories.
---
### c) Suggested Measures
* **Measure of Central Tendency:** The **Mean** is suggested because the data is relatively symmetric (bell-shaped) for both suppliers, making the mean a reliable balance point.
* **Measure of Dispersion:** The **Standard Deviation** is most suitable as it accounts for every data point in the set and is the standard accompaniment to the mean for symmetric distributions.
---
### d) Calculations for Supplier A
To compute these, we use the midpoint ($x$) of each class.
| Weight (g) | Midpoint ($x$) | Freq ($f$) | $f \cdot x$ | $f \cdot x^2$ |
| :--- | :--- | :--- | :--- | :--- |
| 501 - 550 | 525.5 | 5 | 2627.5 | 1,380,751.25 |
| 551 - 600 | 575.5 | 10 | 5755.0 | 3,312,002.50 |
| 601 - 650 | 625.5 | 20 | 12510.0 | 7,825,005.00 |
| 651 - 700 | 675.5 | 12 | 8106.0 | 5,475,603.00 |
| 701 - 750 | 725.5 | 3 | 2176.5 | 1,579,050.75 |
| **Total** | | **50** | **31,175** | **19,572,412.5** |
1. **Mean ($\bar{x}$):** $\frac{\sum fx}{n} = \frac{31175}{50} = \mathbf{623.5\text{g}}$. On average, a carton from A weighs $623.5$g.
2. **Median:** Located at the $25^{\text{th}}$ value (in $601-650$ class). $L + [\frac{n/2 - F}{f}]C = 600.5 + [\frac{25-15}{20}]50 = \mathbf{625.5\text{g}}$. Half the cartons are below $625.5$g.
3. **Mode:** The class with highest frequency ($601-650$). Midpoint = $\mathbf{625.5\text{g}}$. This is the most common weight.
4. **Standard Deviation ($s$):** $\sqrt{\frac{\sum fx^2 - \frac{(\sum fx)^2}{n}}{n-1}} = \sqrt{\frac{19572412.5 - 19437812.5}{49}} = \sqrt{2746.94} \approx \mathbf{52.41\text{g}}$.
---
### e) Quartile Deviation for Supplier B
The Quartile Deviation is $QD = \frac{Q_3 - Q_1}{2}$. For Supplier B:
* **$Q_1$ (12.5th value):** Falls in $601-650$ class. $600.5 + [\frac{12.5-10}{12}]50 \approx \mathbf{610.92\text{g}}$.
* **$Q_3$ (37.5th value):** Falls in $651-700$ class. $650.5 + [\frac{37.5-22}{20}]50 = \mathbf{689.25\text{g}}$.
* **Quartile Deviation:** $\frac{689.25 - 610.92}{2} = \mathbf{39.17\text{g}}$.
---
### f) Decision and Justification
The bakery should choose **Supplier A**.
**Reasoning:** Although Supplier B provides more weight per carton, the prompt emphasizes that **consistency** is vital for the recipe. By looking at the frequency distributions, Supplier A's weights are more tightly clustered around its center ($40$ out of $50$ cartons are between $551$g and $700$g). Supplier B has more spread, particularly with a higher frequency of cartons in the extreme $701-750$g range, which could lead to inconsistent muffin quality.
How do the prices of these two suppliers compare in your scenario?

Question 1162460: A quality-control procedure for testing digital cameras consists of drawing two cameras at random from each lot of 100 without replacing the first camera before drawing the second. If both are defective, the entire lot is rejected. If the lot contains 10 defective cameras, assume that each camera in the lot has an equal chance of being drawn.
1. Are these events dependent or independent? Justify your answer.
2. Use the appropriate multiplication rule to obtain the probability that the entire lot being rejected.
3.If one of the cameras drawn is defective, then another two cameras will be drawn from the second lot.
Find the probability that the two cameras will be drawn from second lot of 100 cameras.
can show the solution teach me how to calculate? thanks.
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
from fractions import Fraction
# Lot details
N = 100
D = 10
ND = 90
# 2. Probability lot is rejected (Both are defective)
# P(D1) = 10/100
# P(D2 | D1) = 9/99
p_rejected = Fraction(10, 100) * Fraction(9, 99)
# 3. Probability exactly one is defective (Condition to draw from second lot)
# P(D1 and ND2) = (10/100) * (90/99)
# P(ND1 and D2) = (90/100) * (10/99)
p_one_defective = (Fraction(10, 100) * Fraction(90, 99)) + (Fraction(90, 100) * Fraction(10, 99))
print(f"{p_rejected=}")
print(f"p_rejected_decimal={float(p_rejected)}")
print(f"{p_one_defective=}")
print(f"p_one_defective_decimal={float(p_one_defective)}")

```
```text?code_stdout&code_event_index=2
p_rejected=Fraction(1, 110)
p_rejected_decimal=0.00909090909090909
p_one_defective=Fraction(2, 11)
p_one_defective_decimal=0.18181818181818182
```
To solve this probability problem, we look at the composition of the lot:
* **Total cameras ($N$):** $100$
* **Defective cameras ($D$):** $10$
* **Non-defective cameras ($ND$):** $90$
---
### 1. Are these events dependent or independent?
These events are **dependent**.
**Justification:**
In probability, two events are independent if the outcome of the first does not change the probability of the second. However, in this case, the cameras are drawn **without replacement**.
* The probability of the first camera being defective is $\frac{10}{100}$.
* If the first camera is defective, there are only $9$ defective cameras left out of $99$ total. The probability of the second being defective changes to $\frac{9}{99}$.
* Since the probability of the second event depends on what happened in the first draw, the events are dependent.
---
### 2. Probability that the entire lot is rejected
The lot is rejected only if **both** cameras drawn are defective ($D_1$ and $D_2$). We use the multiplication rule for dependent events:
$$P(D_1 \cap D_2) = P(D_1) \times P(D_2 | D_1)$$
**Calculation:**
* $P(D_1) = \frac{10}{100}$
* $P(D_2 | D_1) = \frac{9}{99}$
$$P(\text{Rejected}) = \frac{10}{100} \times \frac{9}{99} = \frac{90}{9,900} = \frac{1}{110}$$
$$\text{Probability} \approx \mathbf{0.0091} \text{ (or } 0.91\%)$$
---
### 3. Probability of drawing from the second lot
Based on the problem description, you draw from the second lot if "**one of the cameras drawn is defective**." In the context of the rules provided, this refers to getting **exactly one** defective camera in the first two draws (since getting two would reject the lot immediately).
There are two ways to get exactly one defective camera:
1. The first is defective and the second is good ($D_1 \cap ND_2$).
2. The first is good and the second is defective ($ND_1 \cap D_2$).
**Calculation:**
* **Scenario 1:** $P(D_1 \cap ND_2) = \frac{10}{100} \times \frac{90}{99} = \frac{900}{9,900} = \frac{1}{11}$
* **Scenario 2:** $P(ND_1 \cap D_2) = \frac{90}{100} \times \frac{10}{99} = \frac{900}{9,900} = \frac{1}{11}$
**Total Probability:**
$$P(\text{Exactly One}) = \frac{1}{11} + \frac{1}{11} = \frac{2}{11}$$
$$\text{Probability} \approx \mathbf{0.1818} \text{ (or } 18.18\%)$$
**Summary for the Teacher:** The logic here is to always subtract $1$ from both the numerator and denominator for the second draw if the attribute matches the first draw, or just from the denominator if the attribute is different. This accounts for the "without replacement" rule!

Question 558195: 2x=√(1-3x)
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

2x=√(1-3x) *************** The other person's solutions, are WRONG!! For the RADICAND, 1 - 3x, we have: So, we get: , with <==== Up to this point is CORRECT! (x + 1)(4x - 1) = 0 x + 1 = 0 OR 4x - 1 = 0 Although both solutions , x = - 1 is EXTRANEOUS, and so, ONLY VALID/ACCEPTABLE solution is:

Question 271147: A BOX MEASURES 40CM*50CM*45CM .FIND THE AREA OF THE CARD BOARD REQUIRED TO MAKE 350BOXES
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
All six surfaces, 1 box

For 350 of these boxes, area amount of cardboard

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A BOX MEASURES 40CM*50CM*45CM .FIND THE AREA OF THE CARD BOARD REQUIRED TO MAKE 350BOXES
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The problem is posed in inaccurate way. It should be retold (reformulated) in other way.
        Simply speaking, they want you find the surface area of one single box and then
        multiplied the single area by 350.

        The solution by @mananth is fatally wrong. For example, he writes
        "In a rectangle or square there are 8 faces."
        It is the case when the artificial intelligence works perpendicularly to common sense.

        I came to bring a correct solution.

The surface area of one rectangular box is 2*(ab + ac + bc) = 2*(40*50 + 40*45 + 50*45) = 12100 cm^2. The surface area of 350 boxes is 350 * 12100 = 4235000 cm^2 = 423.5 m^2. ANSWER

Solved correctly.

Question 1032795: Julie wants to build a sidewalk of uniform width around her garden. Her garden is rectangular, and its dimensions are 20 feet by 30 feet. She has enough pavers to cover 900 square feet and wants to use all the pavers. Complete the following statement. Round to the nearest tenth. Julie should make the width of the sidewalk _ _ _ _ _ feet.
Found 2 solutions by timofer, ikleyn:
Answer by timofer(159)   (Show Source):
You can put this solution on YOUR website!
Simplest equation found after some setup and some steps looks like .
Expecting to use quadratic formula solution and take the one with the plus-square root.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Julie wants to build a sidewalk of uniform width around her garden. Her garden is rectangular,
and its dimensions are 20 feet by 30 feet. She has enough pavers to cover 900 square feet
and wants to use all the pavers. Complete the following statement. Round to the nearest tenth.
Julie should make the width of the sidewalk _ _ _ _ _ feet.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the post by @mananth, the value of x2 is determined by @mananth incorrectly
as -15.5.

The correct value for x2 is about -30.

For this problem, the precise value of x2 does not matter, since it is negative.

But I do not think, that your teacher will be glad by seeing wrong value in your solution.

Question 494362: It takes 4 men 14 days to do a certain job. How long should it take 7 men working at the same rate to do the same job?
Found 3 solutions by ikleyn, josgarithmetic, MathTherapy:
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
It takes 4 men 14 days to do a certain job.
How long should it take 7 men working at the same rate to do the same job?
~~~~~~~~~~~~~~~~~~~~

                This can be solved MENTALLY and MOMENTARILY.

(1)   The whole job is   4 * 14 = 56 man-days.

(2)   7 men will make it in   56/7 = 8 days.         ANSWER

Solved.

It is how my teachers taught us solving such problems 65 years ago.

Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
r, work rate of each man
x, the unknown time for the 7 men
1, the amount of jobs

See, each expression on the left equals to 1, so

Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

It takes 4 men 14 days to do a certain job. How long should it take 7 men working at the same rate to do the same job? ************************************ The other person's answer, that it takes 7 men, 24 days to complete a job that 4 men take 14 days to do, makes ABSOLUTELY no sense, especially seeing that both groups are working at the same rate. How can a larger group of men take MORE TIME to a job than a smaller group, when both groups are working at the same rate? RIDICULOUS and NONSENSICAL!! It'll take the 7 men to do the same job that it takes 4 men, 14 days to complete.

Question 1155046: It's a question on indices
Solve for x and y in:
1)3^x-2y*5^y+x
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
It's a question on indices
Solve for x and y in:
1)3^x-2y*5^y+x
~~~~~~~~~~~~~~~~~~~~~~

This post is a compote of words with no mathematical sense.

Every word taken separately makes sense, but taken altogether, they are mathematically non-sensical.

Question 1176752: Using k as the constant of variation, write the equation of variation for each following.
1. The current l varies directly as the electromotive force E and inversely as as the resistance R.
2. The acceleration a of an object varies directly as the force f exerted and inversely as it's mass m.
3. The stiffness s of a beam varies directly as it's depth d and inversely as the square of the length l.
4. The time t required for an elevator to lift a weight varies directly with the distance d through which it is to be lifted and inversely with the power p of the motor.
5. The force f of attraction between two bodies varies directly as the product of their weights w and inversely as the square of the distance d between them.
6. The electrical resistance R of wire varies directly as it's length l and inversely as the square of its diameter d.
7. The acceleration A of a moving objects varies directly as the distance d inversely as the square of the time t.
Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.

Formulation #5 is incorrect.

To understand why it's incorrect, imagine two objects (two masses) in outer space.

There is no weight because there is weightlessness (the weight is zero),
but there is a force of attraction.

The error in the formulation is that instead of the concept of weight, the concept of mass should be used.

Question 1210577: Find the derivative of y=x using the first principle of differenciation
2) y=2x²—x
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!
I assume that in your class what was called the first principle of differentiation is that the derivative of a function is
the limit of when tends to zero.

For or , , ,
but does not depend on ,
so the limit of when tends to zero is and or

For or ,
,
,
, and the limit of when tends to zero is , so

Question 1183412: A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper of radius 70cm. The remaining part is folded to form a cone. Find 1. the vertical angle of the cone 2. The angle of the sector.

Found 4 solutions by n3, n2, CPhill, ikleyn:
Answer by n3(7)   (Show Source):
You can put this solution on YOUR website!
.

        To the managers of this project !
    Attention !!     Attention !!     ATTENTION !!

Today, I had a chance to review the bunch of solutions produced by @CPhill to Math problems at this forum.

               A list of the links follows

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.1182443.html

https://www.algebra.com/algebra/homework/logarithm/logarithm.faq.question.244998.html

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.1182444.html

https://www.algebra.com/algebra/homework/Systems-of-equations/Systems-of-equations.faq.question.1210543.html

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.1182591.html

https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1182886.html

https://www.algebra.com/algebra/homework/playground/test.faq.question.1183412.html

https://www.algebra.com/algebra/homework/playground/test.faq.question.1210545.html

https://www.algebra.com/algebra/homework/Systems-of-equations/Systems-of-equations.faq.question.1210543.html

https://www.algebra.com/algebra/homework/formulas/Geometric_formulas.faq.question.1201106.html

My impressions about his skills as a developer of the AI for solution of school Math problems are at very low level.

This person has no necessary knowledge and understanding Math to work on the AI project
in the area of solution of Math problems.

He doesn't know what kinds of mathematical problems are possible and which ones are not.
He also doesn't know what kinds of solutions to mathematical problems are possible,
and which ones are not and should not be.

Very often he posts nonsense to the forum under the guise of mathematical problems,
and very often he posts nonsense to the forum under the guise of solutions to mathematical problems.

He also has no necessary skills to work in such a project NEITHER as an individual NOR as a member of a team.
For the team work, he has no necessary respect to the work of specialists in this area.

So, if you, the managers, want to continue such a project successfully, you should consider replacing this person
to a more appropriate candidate.

Yours well-wisher, @ikleyn

Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper
of radius 70cm. The remaining part is folded to form a cone. Find
1. the vertical angle of the cone
2. The angle of the sector.
~~~~~~~~~~~~~~~~~~~~~~~~~~~

This post by @CPhill is a copy-paste of the post by @mananth.

They both are very suspicious, because they both contain this phrase
"The area of major arc is the circumference of the base of cone".

As you see this gibberish, you should throw this solution to a garbage box
immediately without any doubts - it does not deserve further consideration.

In addition, the solution in both their posts are incomplete.

In my post (as @ikleyn) at this spot, I gave another, fully correct and complete solution to this problem,
So, you can disregard both posts by @mananth and by @CPhill.

Don't let them cloud your brain with flawed "quasi"-solutions.

Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
Length of arc = +%28%28theta%29%2F360%29%2A+%28pi%29%2Ad%29
length of arc p = 110/360 (2*pi*70)
p= 1540/36 * pi
area of major arc = (140 *pi -(1540/36)*(pi))
= 97.2 *pi
The area of major arc is the circumference of the base of cone
97.2*pi = d *pi
d= 97.2
r = 48.6
Now the radius of the circle becomes the slant height of the cone
height of cone h=+sqrt%28l%5E2-r%5E2%29
height of cone = sqrt%2870%5E2-48.6%5E2%29
height of cone = 50.37
we know radius and height angle of cone can be calculated

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper of radius 70cm.
The remaining part is folded to form a cone. Find 1. the vertical angle of the cone 2. The angle of the sector.

After cutting away the sector of 110°, the remaining pert of the circle is the sector of 360° - 110° = 250°. The length of the arc of the sector of (250°, R = 70 cm) is = = 305.4323611 cm. The radius of the base of the cone 'r' can be defined from = 305.4323611, r = = 48.61111... cm. It is the same as to use equation = , r = = = 48.61111... cm. Now the cone has the slant height of 70 cm and the radius of 48.61111... cm. From the right-angled triangle, for the angle 'a' between the slant height and the cone axis we have sin(a) = = 0.694444... Thus angle 'a' is a = arcsin(0.694444), or about 44°. Vertical angle of the cone is about 2*44° = 88°.

Solved.

Question 1210545: Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including at least 60 of the deluxe kind. The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 . There are 400 hours available for assembling time. The equipment can be used to assemble either kind of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize the company's cost of this special order. You are required as an expert to formulate this problem in a linear programming form and using the graphical method advise the manager on the best product that will enable his firm to minimize it's cost
Found 3 solutions by n2, CPhill, ikleyn:
Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds
that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including
at least 60 of the deluxe kind.
The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas
the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 .
There are 400 hours available for assembling time. The equipment can be used to assemble either kind
of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize
the company's cost of this special order. You are required as an expert to formulate this problem
in a linear programming form and using the graphical method advise the manager on the best product
that will enable his firm to minimize it's cost
~~~~~~~~~~~~~~~~~~~~~~~~

The answer in the post by @CPhill (125 standard cabinets, 60 Deluxe cabinets) does not satisfy
the restriction of 400 hours: 125*4 + 60*5 = 800 hours, which greatly exceeds the restriction of 400 hours.

It also does not satisfy the requirement "at least 200 bathroom cabinets", since 125 + 60 = 185 is less than 200.

So, @CPhill solved DIFFERENT problem from what is posed in the post.

He made it even without explicit announcement/declaration about changing the problem,
which is inappropriate practice and can confuse a reader.

For the correct treatment of the problem, see the post by @ikleyn at this spot,
where it was shown that the problem, as posed in the post, has no solution, at all.

Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
To help Gorimapa Nigeria PLC minimize its costs while meeting the order requirements, we can use **Linear Programming (LP)**. This mathematical approach allows us to find the most efficient combination of standard and deluxe cabinets within the given constraints.
### 1. Problem Formulation
First, we define our variables and the mathematical relationships between them.
**Decision Variables:**
* Let = Number of **Standard** cabinets to produce.
* Let = Number of **Deluxe** cabinets to produce.
**Objective Function:**
The goal is to minimize the total variable cost ():

**Constraints:**
1. **Total Order Quantity:** At least 200 cabinets in total.
*

2. **Deluxe Requirement:** At least 60 deluxe cabinets must be made.
*

3. **Assembly Time:** Total time cannot exceed 800 hours.
*

4. **Non-negativity:** Production cannot be negative.
*

---
### 2. Graphical Method Analysis
To solve this graphically, we treat the inequalities as equations to find the boundary lines.
* **Line 1 (Total Order):** .
* If ; if .

* **Line 2 (Deluxe Min):** .
* A horizontal line at .

* **Line 3 (Time Limit):** .
* If ; if .

#### Identifying the Feasible Region
The feasible region is the area that satisfies all three conditions simultaneously. Looking at the intercepts:
* The **Time Limit** and **Total Order** lines actually meet at the x-axis at .
* However, we must have at least 60 Deluxe cabinets ().
By solving the intersection of and the other constraints:
1. **Intersection of and :** . Point: **(140, 60)**.
2. **Intersection of and :** . Point: **(125, 60)**.
**Wait!** Note that at point (140, 60), the total hours needed would be hours. This exceeds our 800-hour limit.
### 3. Finding the Optimal Solution
The feasible region is a small triangle bounded by the points where these constraints overlap. To minimize cost, we test the corner points of the feasible region:
| Corner Point | Cost Calculation () | Total Cost |
| --- | --- | --- |
| **(125, 60)** | | **#860,000** |
| **(0, 160)** | | #960,000 |
| **(0, 200)** | *Not feasible (Exceeds 800 hours)* | - |
*Note: The point (200, 0) is also not feasible because must be at least 60.*
---
### 4. Expert Advice to the Manager
Based on the linear programming analysis, the best product mix to minimize costs while staying within the 800-hour limit and meeting the order requirements is:
* **Standard Cabinets:** 125 units
* **Deluxe Cabinets:** 60 units
* **Total Production:** 185 units

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds
that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including
at least 60 of the deluxe kind.
The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas
the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 .
There are 400 hours available for assembling time. The equipment can be used to assemble either kind
of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize
the company's cost of this special order. You are required as an expert to formulate this problem
in a linear programming form and using the graphical method advise the manager on the best product
that will enable his firm to minimize it's cost
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This problem, as it is presented in the post, has no solution, at all.

Indeed, to make 200 cabinets, it requires at least 4*200 = 800 hours of work,
counting the minimum required time of 4 hours for a standard cabinet.

It is just more than 400 hours available, so the problem does not have a solution.

You do not need to deploy the heavy artillery of the Linear Programming to get this conclusion.

Question 1210537: The curve surface area of a cylinder is 1.5 times the total surface area of a cone. If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone is 12cm. Calculate the height of the cylinder
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
The curve surface area of a cylinder is 1.5 times the total surface area of a cone.
If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone is 12cm.
Calculate the height of the cylinder
~~~~~~~~~~~~~~~~~~~~~~~~~

The curve surface area of this cylinder is = = = . The slant height of the cone is L = = = = 13 cm. The total surface area of the cone is = + = + = = . Now write the equation = = Simplify it h = 1.5*9 = 13.5. ANSWER. The height of the cone is 13.5 cm.

Solved.

Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
To find the height of the cylinder, we first need to determine the total surface area of the cone and then use the relationship provided to solve for the cylinder's height.
### 1. Find the Dimensions of the Cone
We are given the diameter of the cone is **10 cm**, so the radius () is **5 cm**. The vertical height () is **12 cm**.
To find the Total Surface Area (TSA) of a cone, we first need the **slant height ()**:

### 2. Calculate the Total Surface Area (TSA) of the Cone
The formula for the TSA of a cone is :

---
### 3. Calculate the Curved Surface Area (CSA) of the Cylinder
The problem states that the CSA of the cylinder is **1.5 times** the TSA of the cone:

### 4. Solve for the Height of the Cylinder ()
The formula for the Curved Surface Area of a cylinder is . We know the diameter is 10 cm, so the radius () is **5 cm**:

Now, divide both sides by :

**The height of the cylinder is 13.5 cm.**
Would you like me to calculate the volume of either shape to see how they compare?

Question 1210538: Find the values of x in the range 0°≤x≤360°, which satisfy the equation below
i) sin3xsinx = 2cos2x + 1
ii) 3cotx + tanx - 4 = 0

Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

Dear @CPhill,

Your posts are doing more harm than good to students' learning.

Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
To solve these trigonometric equations, we will use fundamental identities to simplify them into a solvable form.
---
## i)
To solve this, we can use the **Product-to-Sum formula**:

**Step 1: Apply the formula to the left side.**
Let and :

**Step 2: Simplify and rearrange.**
Multiply the entire equation by :

**Step 3: Use the Double Angle formula for .**
Recall that . Here, let :

**Step 4: Factor the quadratic.**
Let :

So, or .
**Step 5: Solve for in the range .**
Since , then .
* **Case 1: **

* **Case 2: **

**Solution (i):**
---
## ii)
**Step 1: Rewrite in terms of .**
Since , the equation becomes:

**Step 2: Form a quadratic equation.**
Multiply through by (noting that ):

**Step 3: Factor the quadratic.**

So, or .
**Step 4: Solve for in the range .**
* **Case 1: **

* **Case 2: **

**Solution (ii):**
---
Would you like me to show you how to verify these solutions using their respective graphs?

Question 1205029: A heavy cube of side 8cm is placed vertically in a cylindrical tank of radius 7cm which contains water.
Calculate the rise in the water level if the original depth of water was:
a) 10 cm
b) 2 cm
Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
A heavy cube of side 8 cm is placed vertically in a cylindrical tank of radius 7 cm which contains water.
Calculate the rise in the water level if the original depth of water was:
    (a)   10 cm
    (b)   2 cm
~~~~~~~~~~~~~~~~~~~~~~~~~~~

(a) In this case, the entire cube is wholly submerged into the water in the tank. The water level rises over the entire base area of the cylindrical tank. The raised water represents the volume of the displaced water in the tank by the solid cube. Use the law of the volume of water conservation. To find the rise for question (a), we should divide the volume of the cube, cm^3 by the area of the base of the cylinder the rise = = = 3.326 cm. ANSWER to question (a). The rise of the water level is 3.326 cm. (b) In this case, the cube is only partly submerged into the water in the tank. The water level rises over the part of the base area of the cylindrical tank. This part of the area where the water rises is the entire area of the base of the tank minus the area of the base of the cube, which is only partially submerged. Use the law of the volume of water conservation. To find the new level of water for question (b), we should divide the volume of the water in the tank, which is cm^3, by the (area of the tank base MINUS area of the cube base) the new level = = 3.4232 cm (rounded). Thus the raise of the water level is 3.4232 - 2 = 1.4232 centimeters. The new level is still lower than the height of the cube, so our calculations make sense. ANSWER to question (b). The rise of the water level is 3.4232 cm.

Solved.

Question 426442: Solve log base 6 of x + log base 6 of (x+16)=2
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

Solve log base 6 of x + log base 6 of (x+16)=2 The other person is WRONG!! The solution set is NOT . Correct x-values are 2 and - 18, of which - 18 should be IGNORED, since it's an EXTRANEOUS solution!! Equation to be solved: , which factors to (x - 2)(x + 18) = 0 You can do the check to CONFIRM!

Question 1206821: The minute-hand of a clock is 6 cm long. How far does the end of the hand travel in 35 minutes?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
The minute-hand of a clock is 6 cm long. How far does the end of the hand travel in 35 minutes?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This my post is written in opposite to the solution by @mananth in his post.

        My goal was to present a short, clear and straightforward solution to the given problem.

A solution is to take = of the circumference of a circle having the radius of 6 cm travel distance is = = 21.99113 cm. Round reasonably to 22 cm. ANSWER

Solved.

Question 1188445: Simplify

Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

Question 446383: If Jill fills a pool with water in 30 min, jack fills a pool with water in 45 min and John fills a pool with water in 1 hour and 30 min. How long will it take if they work together.
Found 3 solutions by math_tutor2020, greenestamps, ikleyn:
Answer by math_tutor2020(3838)   (Show Source):
You can put this solution on YOUR website!

Answer: 15 minutes

Reasoning
1 hr + 30 min = 60 min + 30 min = 90 min

When working alone we have these time durations in minutes only
Jill = 30 min
Jack = 45 min
John = 90 min
The LCM of that set is 90.

Consider a pool with a capacity of 9000 gallons. This hypoethical value can be changed to anything else and the final answer will be the same at the end.
I'm tacking a few zeros onto 90 to get some large capacity, so when we divide it later on we get integer results.

When working alone, Jill fills 9000 gallons in 30 min. Her rate is 9000/30 = 300 gallons per min.
rate = amountDone/time

When working alone, Jack does the same job in 45 min. His rate is 9000/45 = 200 gallons per min.

And when working alone, John's rate is 100 gallons per minute because 9000/90 = 100

The unit rates for each person are then added up. The assumption is that each person doesn't hinder the other when working together.
Eg: John doesn't slow down Jack
Jill + Jack + John = 300+200+100 = 600
They combine to a rate of 600 gallons per minute.

When working together, the pool gets filled in 15 minutes because (9000 gallons)/(600 gallons per min) = 15

Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

Here is an alternative to the standard algebraic method shown by the other tutor. This method works especially well in this particular problem because the numbers are "nice".

The three times for the three people to fill the pool, in minutes, are 30, 45, and 90.

Consider the least common multiple of those times, which is 90 minutes. In 90 minutes, ...
Jill could fill the pool 90/30 = 3 times;
Jack could fill the pool 90/45 = 2 times; and
John could fill the pools 90/90 = 1 time

Therefore, in 90 minutes, the three of them could fill the pool 3+2+1 = 6 times.

And if they can fill the pool 6 times in 90 minutes, the time it takes them to fill the pool once is 90/6 = 15 minutes.

ANSWER: 15 minutes

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
If Jill fills a pool with water in 30 min, Jack fills a pool with water in 45 min and John fills a pool
with water in 1 hour and 30 min. How long will it take if they work together ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        I will solve it in more understandable way than @mananth does it in his post.

Jill makes full job in 30 minutes - hence, Jill makes 1/30 of the job per minute.

Jake makes full job in 45 minutes - hence, Jake makes 1/45 of the job per minute.

John makes full job in 90 minutes - hence, John makes 1/90 of the job per minute.

Working together, they make

         + + = + + = =

of the job per minute.

Hence, it will take 15 minutes for the three participants to complete the job working together.

               Solved - hip-hip-hooray ( ! )

Question 437259: Solve. A boat moves 5 kilometers upstream in the same amount of time it moves 17 kilometers downstream. If the rate of the current is 8 kilometers per hour, find the rate of the boat in still water.
A. 7 1/12 kilometers per hour
B. 3 1/3 kilometers per hour
C. 8 kilometers per hour
D. 14 2/3 kilometers per hour
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

In his solution, @mananth mixed the notions/conceptions "wind" and "current",

so his post partially reminds a soup of words, but with this my notice,

the reader will be able to deal with these details.

But if somebody will integrate his solution into a greater composition, it should be fixed.

Question 437121: 15. sue can shovel snow from her driveway in 30mins. jim can do the same job in 35mins. how long would i ttake sue and ime to shovel teh driveway if they worked together?
16.simplify by removing factors of 1 (7y-14)/2-y
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
Assume each works on the same driveway.
Sue, 30 minutes for the job
Jim, 35 minutes the same job
The two of them working together, driveway per minutes

or looking at reciprocal
minutes per driveway
approximately 16 minutes

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
15. sue can shovel snow from her driveway in 30 mins. jim can do the same job in 35 mins. how long would i ttake sue and ime to shovel teh driveway if they worked together?
~~~~~~~~~~~~~~~~~~~~~~~~

        Calculations in the post by @mananth are incorrect and should be redone (polished).
        I came to make the job accurately and to present it an a way as it SHOULD be done.

Sue makes of the job per minute. Jim makes of the job per minute. Working together, they make + = = = = of the job per minute. So, they need = 16 minutes working together.

Solved correctly and presented properly.

Question 427596: when I open a book two page face me. the sum of the page number is 85. what are the page number ? if the sum is not give, but the product is gives to be 1806, how will your find the page number
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
(a) when I open a book two page face me, the sum of the page number is 85. what are the page number ?
(b) if the sum is not give, but the product is gives to be 1806, how will your find the page number
~~~~~~~~~~~~~~~~~~~~~~~

                Part (a)

Let the page numbers are n and (n+1), two consecutive integer numbers. The problem says that n + (n+1) = 85. Simplify and find 'n' 2n + 1 = 85, 2n = 85 - n1 = 84, n = 84/2 = 42. ANWER. The pages are '42' and '43'.

                Part (b)

This time, the equation is n*(n+1) = 1806. You can solve it by guessing n = 42 or by advanced guessing with reasoning. Alternatively, you can reduce it to quadratic equation n^2 + n - 1806 = 0 and solve it using the quadratic formula or factoring. Or you can evaluate the square root of 1806: = 42.497 approximately, which tells you that n = 42.

Thus, both parts, (a) and (b), are solved.

Question 428529: find the distance between the points (12,8) and (4,2)
1. 100 units
2. 14 units
3. 10 units
4. -10 units
Find the midpoint of the points (3,1) and (7,-5)
1. (1,6)
2. (2,1)
3. (2,-3)
4. (5, -2)

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

Compare my solution with the solution of the other tutor and find differences

Distance between two points x1 y1 x2 y2 d= 12 8 4 2 d= d= d= d= 10 ........................ If the coordinates of A and B are ( x1, y1) and ( x2, y2) respectively, then the midpoint, M, of AB is given by the following formula M = ( 3 , 1 ) ( 7 , -5 ) M = x= ( 3 + 7 )/ 2 y= ( 1 - 5 )/ 2 x= 5 y= -2

Question 729458: A student scored 70 points on a test that had 4 seven- point fill in the blank questions and 24 three point multiple choice questions. If the student answered 1 fill in the blank question , how many multiple choice question did the student answer correctly?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A student scored 70 points on a test that had 4 seven- point fill in the blank questions
and 24 three point multiple choice questions. If the student answered 1 fill in the blank question,
how many multiple choice question did the student answer correctly?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I will solve and answer in one line how many multiple choice questions did the student answer correctly = = = 21. ANSWER

Solved.

The formula is self-explanatory.