Question

An open rectangular box with square base is to be made from 48 ft2 of material.
What dimensions will result in a box with the largest possible volume? Please show work, I'm lost! THanks

Answer 1

base is square with length = width = x, and height = y
volume = x^2*y
minimize f(x,y) = x^2*y
subject to x^2+4xy = 48 (area of square base and 4 "walls" with area w*h)
y = (48-x^2)/4x
f(x,y) = f(x) = x^2((48-x^2)/4x) = x((48-x^2)/4 = (48x-x^3)/4 = 12x - (1/4)x^3
f'(x) = 0 --> 12 - (3/4)x^2 = 0
(3/4)x^2 = 12
x^2 = 16
x= +- 4
-4 does not make sense so
x = 4
so
x^2+4xy = 48 ---> 16+16y = 48 --> y = 2
so max volume is
4*4*2 = 32ft^3

Answer 2

@skdavis93 let me know if you have probs.