Optimization word problem

Question

A rectangular page is to contains 99 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

larger value = ?

smaller value = ?

Very confused, so any help will be much appreciated!

Answer 1

desmos.com/calculator/upilyasqik

Answer 2

Try drawing a diagram.

Suppose the page has width x, length y.

Then we need to take an inch in from both sides and top and bottom, leaving a print area of (x-2) (y-2) = 99

We need to choose x, y so that the area of the paper A = xy is minimised, knowing that (x-2) (y-2) = 99.

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So (x-2) (y-2) = 99

y-2 = (99) / (x-2)

y = 2 + (99 / (x-2)) = (2x-4 + 99 / (x-2)) = = (2x+95) / (x-2)

So A = xy = x (2x+95) / (x-2)

You can use calculus to solve this setting dA/dx = 0

Or plot y = x (2x+95) / (x-2) on Desmos and choose the value of x making y as small as possible.

I just did that, looks like x = 11.95, with a minimum area of 142.799

(y = 142.799 / 11.95).