a rectangular box with a square base has a volume of 64 cubi

a rectangular box with a square base has a volume of 64 cubic feet. determine the dimensions of the box that yield the minimum surface area. what is the minimum surface area?

Solution

V = 64

side of base = x

height = y

x^2 * y = 64

y = 64/x^2

SA = 2x^2 + 4xy

A = 2x^2 + 4x(64/x^2)

A = 2x^2 + 256/x

dA/dx = 4x - 256/x^2 = 0

4x = 256/x^2

x^3 = 64

x = 4

Heght = 64/16 = 4

So, base side length = 4 and height = 4

the SA : 2x^2 + 4xy = 2(4)^2 + 4(4)(4) = 32 + 64 = 96 sq cm