Question

have 50 sq ft of material to make an open top box with a square base.
a) use formula for surface area to express the height h of the box in terms of x.

b) find the dimensions of the box that will produce the maximum volume.

Answer 1

side of square = x
height = h

surface area = x^2 + 4xh

x^2 + 4xh = 50
so,
h = (50 - x^2)/4x

v = hx^2 = (50-x^2)/4x * x^2
v = x(50-x^2)/4
v has a max at x = 5sqrt(2/3) = 4.08

so, a box 4 x 4 x 17/8 will have maximum volume of 34