Optimization Problem

Question

A box is contructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $3 per square foot and the metal for the sides costs $8 per square foot. Find the dimensions that minimize cost if the box has a volume of 40 cubic feet.

Length of base x:

Length of side z:

Answer 1

Hi Kayla,

If you draw a box with a square top and bottom, you will have 6 different surfaces. The top and bottom are both squares, so the area for each will be x * x = x². The four sides will be "z" tall and "x" wide, so the area of each of the side panels is xz.

Since the COST is the thing you want to minimize, the cost of the top and bottom material needs to be multiplied by the area in the top and bottom. The cost of the metal for the sides needs to be multiplied by the area in the sides. So your primary equation is

COST = ($3)(2)x² + ($8)(4)xz

Your constraint (secondary equation) is the information about the volume.

V = (x)(x)(z) = 40 cubic feet.

Solving this for z, find that z = 40/(x²). Substitute this into the primary equation for COST and get

COST = 6x² + (32)(x)(40/x²)

COST = 6x² + 1280/x

Now ready to differentiate the expression for COST, set the derivative =0, and find x, then plug it into z=40/x² and find z.