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#1 2006-02-28 02:09:01
- ganesh
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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.
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#2 2006-02-28 05:48:14
- mathsyperson
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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.
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It wanted to be normal.
#4 2006-02-28 15:46:38
- ganesh
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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.
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#5 2006-03-01 15:20:21
- ganesh
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Re: Calculus
C # 3
Differentiate the following with respect to x.
(i) xx (ii) xsinx
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#6 2006-03-02 03:16:34
- ganesh
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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?
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#8 2006-03-02 14:58:56
- ganesh
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Re: Calculus
Excellent, irspow! 
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#9 2006-03-07 02:04:50
- ganesh
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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?
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#10 2006-03-07 04:29:11
- Ricky
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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..."
#12 2006-03-07 17:25:08
- ganesh
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Re: Calculus
C # 6
If
show that x (dy/dx) = y.
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#13 2006-05-13 05:59:43
- John E. Franklin
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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.
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