A rectangular box with square base and open top is formed from two materials. The base material costs $5 per square foot, while the four sides cost $2 per square foot. If the volume of the box is 10 cubic feet, find the dimensions that minimize the cost.
+130514 V = l * w * h but since the base is square l = w and we'll just call this x
So
V = b * h
V = x^2 * h
10 = x^2 * h and solving for h, we have that
h = 10/x^2
The cost is given by :
Area of the base * cost of material per sq ft + Area of the 4 sides * cost of material per sq ft
So we have
C = x^2 (5) + 4xh (2) and substituting for h , we have
C = 5x^2 + 8x ( 10 / x^2)
C = 5x^2 + 80x^-1 take the derivative of this and set it to 0
C' = 10x - 80x^-2 = 0 multiply both sides by x^2
10x^3 - 80 = 0 divide through by 10
x^3 - 8 = 0 add 8 to both sides
x^3 = 8 take the cube root of both sides
x = cube root (8) = 2 ft ...... since the base is square, this is the length and the width
The height = 10 / [cube root(8)]^2 = 10 / 4 ft = 2.5 ft
And the minimum cost = 5(2)^2 + 80/2 = 20 + 40 = $60
