Question

You need to construct an open-top rectangular box with a square base that must hold a volume of exactly 475 cm3. The material for the base of the box costs 8 cents/cm2 and the material for the sides of the box costs 6 cents/cm2. The dimensions for a box that will minimize the cost of the materials used to construct box are:

Answer 1

The dimensions of the box are:

x = 8,93 cm and h = 5,95 cm

C(min) = 850,69 cents

Explanation:

The volume of the box is:

V = x²*h where x is the side of the square base and h the height

then h = V/ x² ⇒ h = 475 / x²

The total cost of box C is:

C = C₁ + 4*C₂ Where C₁ and C₂ are the costs of the base and one lateral side respectevily

Then cost C = 8*x² + 4* 6*h*x

The cost C as a function of x is

C(x) = 8*x² + (24* 475 /x² )*x

C(x) = 8*x² + 11400/x

Tacking derivatives on both sides of the equation

C´(x) = 16*x - 11400/x²

C´(x) = 0 ⇒ 16*x - 11400/x² = 0

16*x³ = 11400 ⇒ x³ = 11400/16

x³ = 712,5

x = 8,93 cm

and h = 475 / (8,93)² ⇒ h = 5,95 cm

C(min) = 8*79,77 + 4* ( 8,93)*5,95

C(min) = 638,16 + 212,53

C(min) = 850,69 cents

To check if value x = 8,93 would make C(x) minimum we go to the second derivatives

C´´(x) = 16 + 22800/x³ > 0

Then we have a minimum of C at x = 8,93