an open box with a square base is to be constructed from 48 square inches of material. What dimensions will produce a box with maximum volume?

Answer 1

The way I read your question:
You have 48 square inches available in total to make the box.

so let the base be x by x inches and the height y inches
total area available = x^2 + 4xy = 48
y = (48 - x^2)/(4x)

volume = x^2 y
= x^2 (48 - x^2)/(4x)
= 12x - x^3/4

d(volume)/dx = 12 - (3/4)x^2
= 0 for a max of volume
(3/4)x^2 = 12
x^2 = 16
x = √16 = 4

if x = 4, then y = (48 - 16)/16 = 2

the box should have a base of 4 inches by 4 inches and a height of 2 inches