A company determines that its marginal cost, in dollars, for producing $\mathrm{x}$ units of a product is given by
\[
C^{\prime}(x)=3200 x^{-1.8}, \text { where } x \geq 1
\]
Suppose that it were possible for the company to make infinitely many units of this product. What would the total cost be?
The total cost would be $\$ \square$.
(Round to the nearest integer as needed.)
Answer
\(\boxed{4000}\) dollars is the total cost for producing infinitely many units of the product.
Step 1 ：The company's marginal cost for producing x units of a product is given by \(C'(x) = 3200x^{-1.8}\), where \(x \geq 1\).
Step 2 ：If the company could produce infinitely many units of the product, we need to find the total cost. This can be calculated by integrating the marginal cost function from 1 to infinity.
Step 3 ：Performing the integration, we find that the total cost is \(4000\) dollars.
Step 4 ：\(\boxed{4000}\) dollars is the total cost for producing infinitely many units of the product.
