Expert: Socrates - 3/22/2009

Question
A rectangular box with square base and top is to be made to contain 1250 cubic feet. The material for the base cost 35 cents per square foot, for the top 15 cents per square foot, and for the sides 20 cents per square foot. Find the dimensions that will minimize the cost of the box.

Answer
Let x be the length in feet of the edges of the box on the top and bottom. Let y be the height of the box in feet .

1250 = Volume =  (x^2)(y)

so y = 1250/x^2


Area of top = x^2

Area of base = x^2

Area of sides = 4xy = (4x)(1250/x^2) = 5000/x


Cost of box = C(x) = 15x^2 + 35x^2 + (20)(5000/x)


so C(x) = 50x^2 + 100,000/x

Take the derivative

C'(x) = 100x - 100,000/x^2

set this equal to 0 and solve

0 = 100x - 100,000/x^2

0 = x - 1,000/x^2

0 = x^3 - 1,000

x = 10

checking the sign of the derivative , we see that for values of x less than 10 , the derivative is negative , for values of x greater than 10 , the derivative is positive. Thus , the cost is a minimum when x = 10 feet.

y = 1250/x^2 , so , when x = 10 ,  y = 1250 /100 = 12.5


The dimensions are then 10 by 10 by 12.5