# Derivatives to calculate volume and area

Volume. An open box is to be made from a six inch by six inch square piece of material by cutting equal squares from the corners and turning up the sides. Find the volume of the largest box that can be made

Area. A rectangular page is to contain 36 square inches of print . The margins at the top and bottom and on each side are to be 1 and a half inches. Find the dimensions of the paper that will minimize the amount of paper used.

Area. A rectangular page is to contain 30 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page such that the least amount of paper is used.

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#### Solution Preview

a)

V = x^2(6-x) = 6x^2 - x^3

dV/dx = = 12x-3x^2 = 0

x = 4

V = 32

b)

A = xy

restriction: (x-3)(y-3) = 36

x = 36/(y-3) + 3

A = ...

#### Solution Summary

This shows three applications of derivatives to calculate volume and area.