Answer:
The dimensions of the box are:
x = 8,93 cm and h = 5,95 cm
C(min) = 850,69 cents
Step-by-step 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
