We want to build a box whose base is square, has no top and will enclose 100 m^3. Determine the dimensions of the box that will minimize the amount of material needed to construct the box.

Answer 1

V= 100 = hx^2

h=100/x^2

material needed = surface area =bottom area + 4 times side area = x^2 + 4hx= x^2 + 4(100/x^2)x = x^2 + 400/x

SA = x^2 + 400/x

SA' = 2x - -400/x^2 = 0

2x^3 = 400

x^3 = 200

x = cube root of 200= about 5.85 m

h = 100/(5.85)^2 = 100/34.22 = about 2.92 m

dimensions are about 5.85 by 5.85 by 2.92 m