# optimal dimensions

Applications of derivatives

You are planning to make an open rectangular box from an 8 by 15 inch piece of cardboard by cutting congruent squares from the corners and folding up the sides. what are the dimensions of the box of largest volume you can make this way, and what is its volume.

Let a be the height of the folded sides. The congruent squares that you cut away will have area of a^2 each. The volume of the box will be
V = (8-2a)(15 - 2a) a
= a (120 + 2a^2 -46 a)
Differentiate V with respect to a and set the derivative equal to zero, and solve for a.

= a (120 + 2a^2 -46 a)
it should be
= a (120 + 4a^2 -46 a)

The volume comes out to be 2450/27 in^3 and a = to 5/3

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