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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

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

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

C # 3

Differentiate the following with respect to x.

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

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
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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?

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

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

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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

Excellent, irspow!

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

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?

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

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

"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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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

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

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 46,424

C # 6

If

show that x (dy/dx) = y.

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

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

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

*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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