Question

an open top box is to be made of sheet of paper that coast 1 cent per square inch.The box has a square bottom which is reinforce with a double layer and volume of 216 cubic inches. Find the dimension of box that minimize the total cost of paper

Answer 1

dimensions to minimize cost is 6 inches x 6 inches x 6 inches

Explanation:

Since the box has a square bottom, then it means length and width are the same value. Let the length and width be x. Let the depth by y.

Thus;

Volume is; V = x²y

We are given volume as 216 in³

Thus, V = 216

x²y = 216

y = 216/x²

Surface area of box will be;

S = 2x² + 4xy

Since box is to be made of sheet of paper that coast 1 cent per square inch.

It's means per Sq.m is $0.01

Thus;

C(x) = 2 × 0.01(x²) + 4 × 0.01(xy)

C = 0.02x² + 0.04xy

Put 216/x² for y;

C = 0.02x² + 0.04x(216/x²)

C = 0.02x² + 8.64/x

dC/dx = 0.04x - 8.64/x²

At dC/dx = 0, cost is minimum

Thus;

0.04x - 8.64/x² = 0

0.04x = 8.64/x²

x³ = 8.64/0.04

x³ = 216

x = 6

From y = 216/x²

y = 216/6²

y = 6

Thus,dimensions to minimize cost is 6 inches x 6 inches x 6 inches