An open box (has a bottom but no top) needs to be constructed which has a volume of 500 cu. in. The bottom of the box is to be a square. Let x be one of the sides of the bottom. Express the area of the four sides and bottom of box as a function of x alone. What value of x will make the total area of the 4 sides and the bottom of the box as small as possible? (that is, gives the box that uses the least amount of material to build and has a volume of 500).

Question

An open box (has a bottom but no top) needs to be constructed which has a volume of 500 cu. in. The bottom of the box is to be a square. Let x be one of the sides of the bottom. Express the area of the four sides and bottom of box as a function of x alone. What value of x will make the total area of the 4 sides and the bottom of the box as small as possible? (that is, gives the box that uses the least amount  of material to build and has a volume of 500).

Please help and explain. I have an exam shortly and a similar question will be on it.

Reiny · Answer

If the base is x by x, let the height be y
Volume = (x^2)y
x^2y = 500
y = 500/x^2

surface area = x^2 + 4xy
= x^2 + 4x(500/x^2 = x^2 + 2000/x

d(surface area)/dx = 2x - 2000/x^2
= 0 for a max/min of surface area

2x - 2000/x^2 = 0
x^3 = 1000
x = 10