Question

A rectangular box with a volume of 1088 ftcubed is to be constructed with a square base and top. The cost per square foot for the bottom is 15cents​, for the top is 10cents​, and for the sides is 2.5cents. What dimensions will minimize the​ cost?

What are the dimensions of the box?A. The length of one side of the base is ____ ft?B. The hieght of the box is ____ft?

Answer 1

x = 11,09 ft

h = 8,85 ft

Explanation:

Box with square base and top

Let x b a side of the square

A₁ (area of base) A₁ = x²

area of the top is equal to area of base A₂ = x²

Cost of area A₁

C₁ = 15 * x²

Cost of area A₂

C₂ = 10 * x²

Area of the side of height h

A₃ = 2*π*x*h V = 1088 = x²*h h = 1088/x²

A₃ = 2*π*x* (1088)/x² A₃ = 6832,64/x

Cost of A₃

C₃ = 2,5*4 *6832,64/x C₃ = 68326,4/x

Total cost C

C(x) = 15x² + 10x² + 68326,4/x

Taking derivatives on both sides of the equation

C´(x) = 30 x + 20 x - 68326,4/x²

C´(x) = 50 x - 68326,4 /x² C´(x) = 0

50 x - 68326,4 /x² = 0

50x³ = 68326,4

x³ = 1366,53

x = 11,09 ft and the height h = 1088/x²

h = 8,85 ft