# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

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## #1 2006-02-27 03:09:01

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Calculus

C # 1

An advertisement is to contain 75 sq. cms. of printed area. There is a 2 cm margin at the bottom, 3 cm margin on each side and no margin at the top. Find the dimensions of the smallest possible paper.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #2 2006-02-27 06:48:14

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Calculus

If the printed width is x, then the printed height must be (75/x), so that the total area is 75.

There is a 3cm margin on each side, so 6 must be added to the width. Similarly, there is a 2cm margin at the bottom, so 2 must be added to the height.

Therefore, the equation that needs to be minimised is (x + 6)(75/x + 2).

Multiplying out of brackets makes this become 75 + 450/x + 2x + 12.

Differentiating gives 2 - 450/x².

The paper is smallest when the differential is equal to 0.

2 - 450/x² = 0
450/x² = 2
450 = 2x²
x² = 225
x = 15cm

That also had a negative answer, but it was discarded because we are dealing with length.

Anyway, using this value with the original equation shows that the smallest possible paper is 21 * 7 = 147cm² big.

Why did the vector cross the road?
It wanted to be normal.

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## #3 2006-02-27 16:39:06

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #4 2006-02-27 16:46:38

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 2

The radius of a spherical balloon is increasing at the rate of 5 cm per second when inflated by pumping air. Find the rate of increase of (i) its surface area and (ii) its volume, when the radius is 4 cm.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #5 2006-02-28 16:20:21

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 3

Differentiate the following with respect to x.

(i) x[sup]x[/sup]       (ii) x[sup]sinx[/sup]

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #6 2006-03-01 04:16:34

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 4

The volume of a cube is increasing at the rate of 7 cubic centimetres per second. How fast is the surface area increasing when the length of the edge is 12cm?

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #7 2006-03-01 12:56:52

irspow
Member
Registered: 2005-11-24
Posts: 1,055

### Re: Calculus

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

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## #8 2006-03-01 15:58:56

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

Excellent, irspow!

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #9 2006-03-06 03:04:50

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 5

The time of a complete oscillation of a simple pendulum of length l is given by the relation T = 2 π √(l/g) where g is a constant. By what percent should the length be changed in order to correct a loss of 2 minutes per day?

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #10 2006-03-06 05:29:11

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Calculus

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #11 2006-03-06 16:48:18

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #12 2006-03-06 18:25:08

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 6

If

show that x (dy/dx) = y.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #13 2006-05-12 07:59:43

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: Calculus

On C5, please clarify the two minute variation.
For example, 1.) when a perfect clock strikes midnight, this slow clock reads precisely 23 hr 58 minutes,
or, 2.) when this slow clock reaches midnight, the perfect clock reads 0 hr 02 minutes, which is ever so slightly different.
#1 is slower by factor of (23 58/60)/24, while #2 is slower by factor of 24/(24 2/60).
#1 is less perfect a clock than #2 because it is analogous to 2/3's and 3/4's or  9/10's and 10/11's.

igloo myrtilles fourmis

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## #14 2015-09-05 07:17:41

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,434
Website

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## #15 2021-04-02 03:01:03

666 bro
Member
From: Flatland
Registered: 2019-04-26
Posts: 706

### Re: Calculus

Last edited by 666 bro (2021-04-02 03:03:02)

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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The text passage contains a series of calculus problems and solutions. It starts with a user posting a calculus problem, followed by another user providing a solution, and then the original user posting more problems. The problems are related to optimization, rates of change, and differentiation.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2006-02-27 03:09:01

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Calculus

C # 1

An advertisement is to contain 75 sq. cms. of printed area. There is a 2 cm margin at the bottom, 3 cm margin on each side and no margin at the top. Find the dimensions of the smallest possible paper.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #2 2006-02-27 06:48:14

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Calculus

If the printed width is x, then the printed height must be (75/x), so that the total area is 75.

There is a 3cm margin on each side, so 6 must be added to the width. Similarly, there is a 2cm margin at the bottom, so 2 must be added to the height.

Therefore, the equation that needs to be minimised is (x + 6)(75/x + 2).

Multiplying out of brackets makes this become 75 + 450/x + 2x + 12.

Differentiating gives 2 - 450/x².

The paper is smallest when the differential is equal to 0.

2 - 450/x² = 0
450/x² = 2
450 = 2x²
x² = 225
x = 15cm

That also had a negative answer, but it was discarded because we are dealing with length.

Anyway, using this value with the original equation shows that the smallest possible paper is 21 * 7 = 147cm² big.

Why did the vector cross the road?
It wanted to be normal.

Offline

## #3 2006-02-27 16:39:06

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #4 2006-02-27 16:46:38

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 2

The radius of a spherical balloon is increasing at the rate of 5 cm per second when inflated by pumping air. Find the rate of increase of (i) its surface area and (ii) its volume, when the radius is 4 cm.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #5 2006-02-28 16:20:21

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 3

Differentiate the following with respect to x.

(i) x[sup]x[/sup]       (ii) x[sup]sinx[/sup]

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #6 2006-03-01 04:16:34

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 4

The volume of a cube is increasing at the rate of 7 cubic centimetres per second. How fast is the surface area increasing when the length of the edge is 12cm?

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #7 2006-03-01 12:56:52

irspow
Member
Registered: 2005-11-24
Posts: 1,055

### Re: Calculus

I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

Offline

## #8 2006-03-01 15:58:56

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

Excellent, irspow!

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #9 2006-03-06 03:04:50

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 5

The time of a complete oscillation of a simple pendulum of length l is given by the relation T = 2 π √(l/g) where g is a constant. By what percent should the length be changed in order to correct a loss of 2 minutes per day?

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #10 2006-03-06 05:29:11

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Calculus

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

## #11 2006-03-06 16:48:18

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #12 2006-03-06 18:25:08

Jai Ganesh
Registered: 2005-06-28
Posts: 47,071

### Re: Calculus

C # 6

If

show that x (dy/dx) = y.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #13 2006-05-12 07:59:43

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: Calculus

On C5, please clarify the two minute variation.
For example, 1.) when a perfect clock strikes midnight, this slow clock reads precisely 23 hr 58 minutes,
or, 2.) when this slow clock reaches midnight, the perfect clock reads 0 hr 02 minutes, which is ever so slightly different.
#1 is slower by factor of (23 58/60)/24, while #2 is slower by factor of 24/(24 2/60).
#1 is less perfect a clock than #2 because it is analogous to 2/3's and 3/4's or  9/10's and 10/11's.

igloo myrtilles fourmis

Offline

## #14 2015-09-05 07:17:41

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,434
Website

Offline

## #15 2021-04-02 03:01:03

666 bro
Member
From: Flatland
Registered: 2019-04-26
Posts: 706

### Re: Calculus

Last edited by 666 bro (2021-04-02 03:03:02)

"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

Offline
user
The text passage contains a series of calculus problems and solutions. It starts with a user posting a calculus problem, followed by another user providing a solution, and then the original user posting more problems. The problems are related to optimization, rates of change, and differentiation.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS