Answer:
C(min) = 120 $
Step-by-step explanation:
Let x be the side of the square Then
Area of the base b equal to area of the top
A(b) = A(t) = x²
Cost (base + top ) C₁ = 2 * 10 * x² C₁ = 20*x²
Sides area are equal
x*h by 4 and V of the box is V = 4 = x²*h
then h = 4/x²
Area of all sides = 4* x* 4/x² 16/x
And cost of sides area is
C₂ = 20*16/x
Total cost ($)
C = C₁ + C₂
C(x) = 20 x² + 320/x
Taking derivatives on both sides of the equation
C´(x) = 40 *x - 320/x²
C´(x) = 0 40 *x - 320/x² = 0 x³ - 8 = 0
x = 2
And minimun cost
C(min) = 20 * (2) + 320/(2)²
C(min) = 40 + 80
C(min) = 120 $
