Answer:
x = 1,5 cm
h = 6 cm
C(min) = 135 $
Step-by-step explanation:
Volume of the box is :
V(b) = 13,5 cm³
Aea of the top is equal to area of the base,
Let call " x " side of the base then as it is square area is A₁ = x²
Sides areas are 4 each one equal to x * h (where h is the high of the box)
And volume of the box is 13,5 cm³ = x²*h
Then h = 13,5/x²
Side area is : A₂ = x* 13,5/x²
A(b) = A₁ + A₂
Total area of the box as functon of x is:
A(x) = 2*x² + 4* 13,5 / x
And finally cost of the box is
C(x) = 10*2*x² + 2.50*4*13.5/x
C(x) = 20*x² + 135/x
Taking derivatives on both sides of the equation:
C´(x) = 40*x - 135*/x²
C´(x) = 0 ⇒ 40*x - 135*/x² = 0 ⇒ 40*x³ = 135
x³ = 3.375
x = 1,5 cm
And h = 13,5/x² ⇒ h = 13,5/ (1,5)²
h = 6 cm
C(min) = 20*x² + 135/x
C(min) = 45 + 90
C(min) = 135 $
