What are the dimensions of an open there is no top rectangul

What are the dimensions of an open (there is no top) rectangular box that has a square base, a capacity of 10 in.^3, and is constructed using the least amount of the material? (Use your calculator, do not expect the answer to be an integer)

Solution

Let h represents height and w represents width and l is the length
Capacity is given by:
Volume=lwh
10 = hw²
Solve for h:
h = 10/w²

The material used will be the areas of the four sides plus the area of the base:
a = 4hw + w²
Substitute for h:
a = 4w*10/w² + w² = 40/w + w²
Take the derivative da/dw, set it equal to zero and solve:
a\' = -40/w² + 2w
0 = 2w - 40/w²
40/w² = 2w
40 = 2w³
20 = w³
w = 2.7144 inches this is also our length as its a square base
10=h(2.7144)^2
10/(2.7144)^2=h
h=3.3840 inches