Answer:
x= 5,77 *∛ K₂ / ( K₁ + K₅ ) the side f the square bottom-top
h = 2,88/ [∛ K₂ / ( K₁ + K₅ ) ]² heigh of the box
Step-by-step explanation:
Data from problem statement only gives one relation between dimensions of the box, we need two, in order to express surface area as a function of just one variable. In such a case we must assume the box is of square bottom and top.
Then
Area of the bottom A(b) = x² ⇒ C(b) = K₁*x²
Area of the top A(t) = x² ⇒ C(t) = K₅*x²
Total lateral area ( 4 sides) A(l) = 4*x*h ⇒ C(l) = 4*K₂*x*h
V(bx) = 96 in³
V(bx) = x²*h = 96
h = 96/x²
Then total cost as a function of x
C(x) = K₁*x² + K₅*x² + 4*K₂*(96)/x
Taking derivatives on both sides of the equation
C´(x) = 2*K₁*x + 2*K₅*x - 384*K₂/x²
C´(x) = 0 ⇒ 2*K₁*x + 2*K₅*x - 384*K₂/x² = 0
2*K₁*x³ + 2*K₅*x³ = 384*K₂ or K₁*x³ + k₅*x³ = 192*K₂
x³ ( K₁ + K₅ ) = 192*K₂
x = ∛192*K₂ / ( K₁ + K₅ )
x= 5,77 *∛ K₂ / ( K₁ + K₅ )
If we obtain the second derivative
C´´(x) = 2*K₁ + 2*K₅ - (-2*x)*384*K₂/x⁴
C´´(x) = 2*K₁ + 2*K₅ + 768*K₂/x³
As x can not be negative the expression C´´ wil be C´´> 0
then we have a minimum for the function for x = 5,77 *∛ K₂ / ( K₁ + K₅ )
and h = 96 / [ 5,77 *∛ K₂ / ( K₁ + K₅ )]²
h = 2,88/ [∛ K₂ / ( K₁ + K₅ ) ]²
