The packaging department for Czerka Corporation needs to build a box with a square base and open top that has a volume of 32 m3 .

Question

The material for the bottom of the box costs $8 per square meter, and the material for the sides costs $1 per square meter. Find the cost of the least expensive box.

Answer 1

Cost = C = 8x^2 + 4xh where x = the side of the square bottom and height = height

Volume = 32 = hx^2

h= 32/x^2

C= 8x^2 + 4x(32/x^2)

C' = 16x -128/x^2 = 0

16x^3 - 128 = 0

x^3 -8 = 0

(x-2)=0

x= 2 meters

h= 32/2^2 = 8 meters

minimum Cost = 8(4) + 4(2)(8) = 32 + 64 = $96

x^3-8 = (x-8)(x^2+2x+4) =0

x^2 +2x +4 =0

x = -2/2 + or - (1/2)sqr(4-16) = 2 imaginary solutions that can be ignored