A clothing company determines that its marginal cost, in dollars per dress, is given by the function below. Find the total cost of producing the first 220 dresses, disregarding any fixed costs.
\[
C^{\prime}(x)=-\frac{4}{25} x+55, \text { for } x \leq 450
\]
The total cost is $\$$
(Round to the nearest cent as needed.)
Answer
Final Answer: The total cost of producing the first 220 dresses is \(\boxed{$8228.00}\).
Step 1 ：The marginal cost function, \(C'(x)\), gives the cost to produce one additional dress. To find the total cost of producing the first 220 dresses, we need to integrate the marginal cost function from 0 to 220. This will give us the total cost of producing 220 dresses.
Step 2 ：Given the marginal cost function \(C'(x) = 55 - 0.16x\), we can calculate the total cost.
Step 3 ：The total cost of producing the first 220 dresses is calculated to be $8228.00.
Step 4 ：Final Answer: The total cost of producing the first 220 dresses is \(\boxed{$8228.00}\).
