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Calculus/Calculus - Min/Max

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Expert: - 3/30/2009


Question
You are designing a rectangular poster to contain 256 in^2 of printing with a 3" margin at the top and bottim and a 2 " margin on each side.  What overall dimensions will minimize the amount of paper used?  Round to nearest whole inch

Answer
It has width W and height H.  WH=256.
With the perimeter, it's (W+2+2)(H+3+3) = (W+4)(H+6).
We can see from the first line of this answer that H=256/W.
Putting this in the equation (W+4)(H+6) for H gives us
(W + 4)(6 + 256/W).

The area, when multiplied out, is 6W + 280 + 1024/W.
We will say that this is A(W).
A'(W) = 6 - 1024/W².

Note that its a concave down parabola { that's what the minus in front of the W² does }, setting the derviatve to 0 will find the maximum.

When we say that 6 - 1024/W² = 0, we can change it to 6W² = 1024.
This changes to W² = 1024/6, so W = √(512/3).

Now 512 / 3 = 170 2/3, so the squareroot is barely over 13.
This is because 13² = 169.

The answer would be W = 13 and H = 20 { since H = 256/13;
if that 256 were a 260, the answer would be 20 }.

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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