Find the dimensions of the open top rectangular box with a square base that has a volume of 4000 cubic inches and has the smallest possible surface area

Question

(A) list the quantities that are varying, the quantity that is constant, and what you are trying to find

(b) define all the variables for the problem and use them to describe a formula for the quantity to be optimized and a formula for the constraint

(c) solve the problem

Answer 1

An open top rectangular box has 4 sides and a base. The base area is x² where x is a side of the base.

Each of the 4 sides has area hx where h=height. Volume = 4000=hx² or h=4000/x² That means each of the sides' area = hx = 4000x/x² = 4000/x

Adding up the 4 areas plus the base area is 4(4000)/x + x² = 16000/x+x² Take the first derivative of that

2x - 16000/x² and set it equal to zero and solve for x

2x³-16000 = 0

x³=8000

x=20 inches

The box is 20x20x 10 inches

h=4000/x²=4000/400=10

20x20x10 inches minimizes the surface area of the box, given 4000 cubic inches for the volume