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"A printed page is being designed to contain up to 96 square inches of printed material. The margins are 1” on the left and right and 1.5” on the top and bottom. Find the outer dimensions of the page of least area which will meet the stated requirements."

I'm confused about this question. It looks like the page of least area will have dimensions 2” by 3” because the only factor limiting the minimum is the margins; the fact that it's designed to contain "up to 96 square inches" doesn't affect the minimum area. Am I reading something incorrectly, or is the problem worded incorrectly?

Waffle
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2 Answers2

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They are talking about the size of the entire page.

To make calculations easier. Lets say the page requirement was 36 square inches. Two possible configurations would be 6 by 6 or 12 by 3.

Now if we add the margins, we would have 8 by 9 or 15 by 5.

See the difference?

We have, $$ A(l) = 96 + 3(l+2) + 2(3 + \frac{96}{l}). $$

See the following drawing.

enter image description here

  • So the actual area of the page is exactly 96 square inches, and the question asks to minimize the area between the margins? – Waffle Sep 28 '14 at 04:38
  • The page area requirement is without the margins included. So when you add the margins, it changes the area. Depending on the dimensions, the area changes by differing amounts. –  Sep 28 '14 at 04:39
  • So the answer is 12” by 8”. – Waffle Sep 28 '14 at 04:45
  • Yes, that's correct. –  Sep 28 '14 at 05:00
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Here is a diagram.

diagram

With these kind of physical problems the general approach is:

  • First draw a diagram, and define your variables on it
  • Identify the constraints: $(Y-3)(X-2) \ge 96$
  • Identify what you are trying to optimise: $argmin_{(X,Y)} XY$

Then you can solve the problem. In this case a early step, is to realise that for minimising area, the then $(Y-3)(X-2) = 96$ This will allow you to express X in terms of Y (or visa versa) and so reduce it down to a single variable problem.

Drawing a diagram normally will clear things up alot.

Frames Catherine White
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