Expert: Paul Klarreich - 10/23/2006

Question
A rectangular box is to have a square base and a volume of 24 cubic feet. Te
material for the top and the base costs of $2.00 per square foot, while the
material for the sides costs $2.25 per square foot.

Find a formula for the cost of the box in terms of the dimensions of the box.
Determine the dimensions of the box that minimize the construction costs..

Answer
Questioner:  RAsheedah Brown
Category:  Calculus
 
Subject:  Calculus problem
Question:  A rectangular box is to have a square base and a volume of 24 cubic feet. The material for the top and the base costs of $2.00 per square foot, while the
material for the sides costs $2.25 per square foot.

Find a formula for the cost of the box in terms of the dimensions of the box.
Determine the dimensions of the box that minimize the construction costs..
...........................................................
Hi, Rasheedah,

In doing these, the scheme is:

Determine the variables -- give them names.
Determine the quantity to be 'optimized'.
Express that quantity in terms of the variables.

IF there are more than one, use some relationship (some 'constraint') to eliminate all but one of the variables.

THEN you differentiate, set the derivative equal to zero, and solve.

Interpret your answer, making sure it solves the problem.

The variables are

x = the width and length of the base (because it is a square)
z = the height of the box.
C = the cost of the materials.

C is to be minimized.

C is the sum of the costs of the top, bottom, and sides.

Top:  x^2 * $2.00 = 2x^2
Bottom: Also 2x^2.
Each side: Area is  xz, and it costs $2.25/sq foot. So each side costs 2.25xz, and there are four of these, so the cost of the side stuff is  9xy.

Ready to go:

C = 2x^2 + 9xy.

Since C is expressed in terms of two variables, use the constraint:

Volume = 24.

x^2 z = 24,  so  z = 24/x^2

C = 2x^2 + 9x(24/x^2) =  2x^2 + 216/x = 2x^2 + 216x^-1

Ready to differentiate:

dC/dx = 4x - 216/x^2

Set that equal to zero:

4x - 216/x^2 = 0

4x = 216/x^2
4x^3 = 216
x^3 = 54

x = cuberoot(54) = cuberoot(27)cuberoot(2)

x = 3 cuberoot(2)

AND:

z = 24/x^2 = 24/(9 cuberoot(4)) = 8/(3 cuberoot(4))