Question

A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs $0.35 per square foot, the material for the sides costs $0.10 per square foot, and the material for the top costs $0.15 per square foot, determine the dimensions of the box that can be constructed at minimum cost.
A. length ft
B. width ft
C. height ft.

Answer 1

Answer:

Length = 2 ft

Width = 2 ft

Height = 5 ft

Step-by-step explanation:

Let the square base of the box has one side = x ft

Therefore, area of the base = x² ft²

Cost of the material to prepare the base = $0.35 per square feet

Cost to prepare the base = $0.35x²

Let the height of the box = y ft

Then the volume of the box = x²y ft³ = 20

$y=\frac{20}{x^{3} }$ -----(1)

Cost of the material for the sides = $0.10 per square feet

Area of the sides = 4xy

Cost to prepare the sides of the box = $0.10 × 4xy

                                                             = $0.40xy

Cost of the material to prepare the top = $0.15 per square feet

Cost to prepare the top = $0.15x²

Total cost of the box = 0.35x² + 0.40xy + 0.15x²

From equation (1),

Total cost $C=0.35x^{2}+(0.40x)\times \frac{20}{x^{2} }+0.15x^{2}$

$C=0.35x^{2}+\frac{8}{x}+0.15x^{2}$

$C=0.5x^{2}+\frac{8}{x}$

Now we take the derivative of C with respect to x and equate it to zero,

$C'=0.5(2x)-\frac{8}{x^{2}}$ = 0

$x-\frac{8}{x^{2}}=0$

$x=\frac{8}{x^{2} }$

$x^{3}=8$

x = 2 ft.

From equation (1),

$(2)^{2}y=20$

4y = 20

y = 5 ft

Therefore, Length and width of the box should be 2 ft and height of the box should be 5 ft for the minimum cost to construct the rectangular box.