Question

a rectangular box with a square base is to have a volume of 20 cubic feet. the material for the base costs $5.5 per square foot. the material for the sides costs $3.5 per square foot. the material for the top costs $4 per square foot. determine the equation of the cost function in terms of ( x ), where ( x ) is the length of the side of the base. what would be the total material cost of the box for the base length of 5 feet?

\\( \circ \\) a. cost function ( c(x) = \frac{8.5x^3 + 280}{x} ), material cost $268.5.

\\( \circ \\) b. cost function ( c(x) = \frac{9.5x^3 + 280}{x} ), material cost $293.5.

\\( \circ \\) c. cost function ( c(x) = \frac{7.5x^3 + 440}{x} ), material cost $255.5.

\\( \circ \\) d. cost function ( c(x) = \frac{9.5x^3 + 280}{x} ), material cost $285.2.

Explanation:

Step1: Define variables and volume

Let $x$ = side length of square base, $h$ = height of box.
Volume formula: $V = x^2h = 20$, so $h = \frac{20}{x^2}$

Step2: Calculate base and top cost

Base cost: $5.5x^2$, Top cost: $4x^2$. Combined: $5.5x^2 + 4x^2 = 9.5x^2$

Step3: Calculate side surface area cost

4 sides, each area $xh$. Total side area: $4xh$.
Side cost: $3.5 \times 4xh = 14xh$
Substitute $h = \frac{20}{x^2}$: $14x \times \frac{20}{x^2} = \frac{280}{x}$

Step4: Build total cost function

Total cost $C(x) = 9.5x^2 + \frac{280}{x} = \frac{9.5x^3 + 280}{x}$

Step5: Compute cost at $x=5$

Substitute $x=5$ into $C(x)$:
$C(5) = \frac{9.5(5)^3 + 280}{5} = \frac{9.5 \times 125 + 280}{5} = \frac{1187.5 + 280}{5} = \frac{1467.5}{5} = 293.5$

Answer:

b. Cost function $C(x) = \frac{9.5x^3 + 280}{x}$, Material cost $293.5$.