Questions: A consulting firm, using statistical methods, provided a veterinary clinic with the cost equation C(x) = 0.00047(x-200)^3+241,698 100 ≤ x ≤ 1,000 where C(x) is the cost in dollars for handling x cases per month. The average cost per case is given by C̄(x) = C(x) / x. Complete parts (A) through (C). (A) Write the equation for the average cost function C̄. C̄(x) =

A consulting firm, using statistical methods, provided a veterinary clinic with the cost equation

C(x) = 0.00047(x-200)^3+241,698

100 ≤ x ≤ 1,000

where C(x) is the cost in dollars for handling x cases per month. The average cost per case is given by C̄(x) = C(x) / x. Complete parts (A) through (C).

(A) Write the equation for the average cost function C̄.

C̄(x) =

Transcript text: A consulting firm, using statistical methods, provided a veterinary clinic with the cost equation \[ \begin{aligned} C(x)= & 0.00047(x-200)^{3}+241,698 \\ & 100 \leq x \leq 1,000 \end{aligned} \] where $C(x)$ is the cost in dollars for handling $x$ cases per month. The average cost per case is given by $\bar{C}(x)=\frac{C(x)}{x}$. Complete parts $(A)$ through $(C)$. (A) Write the equation for the average cost function $\overline{\mathrm{C}}$. \[ \overline{\mathrm{C}}(\mathrm{x})= \]

Solution

Solution Steps

To find the average cost function $\overline{C}(x)$, we need to divide the total cost function $C(x)$ by the number of cases $x$. Given the cost function $C(x) = 0.00047(x-200)^3 + 241,698$, we can express the average cost function as $\overline{C}(x) = \frac{C(x)}{x}$.

Solution Approach

Define the cost function $C(x)$.
Divide $C(x)$ by $x$ to get the average cost function $\overline{C}(x)$.

Step 1: Define the Cost Function $C(x)$

The cost function is given by: \[ C(x) = 0.00047(x - 200)^3 + 241698 \]

Step 2: Define the Average Cost Function $\overline{C}(x)$

The average cost function $\overline{C}(x)$ is obtained by dividing the total cost function $C(x)$ by the number of cases $x$: \[ \overline{C}(x) = \frac{C(x)}{x} = \frac{0.00047(x - 200)^3 + 241698}{x} \]

Step 3: Calculate the Average Cost for $x = 500$

Substitute $x = 500$ into the average cost function: \[ \overline{C}(500) = \frac{0.00047(500 - 200)^3 + 241698}{500} \]

Step 4: Simplify the Expression

First, calculate the term inside the parentheses: \[ 500 - 200 = 300 \] Then, raise it to the power of 3: \[ 300^3 = 27000000 \] Next, multiply by the coefficient: \[ 0.00047 \times 27000000 = 12690 \] Add the constant term: \[ 12690 + 241698 = 254388 \] Finally, divide by $500$: \[ \overline{C}(500) = \frac{254388}{500} = 508.776 \]