The marginal average cost of producing $x$ digital sports watches is given by the function $\bar{C}^{\prime}(x)$, where $\bar{C}(x)$ is the average cost in dollars.
\[
\bar{C}^{\prime}(x)=-\frac{1,800}{x^{2}}, \quad \bar{C}(100)=21
\]
Find the average cost function and the cost function. What are the fixed costs?
Answer
The fixed costs are the costs that do not change with the quantity of output, which is the constant term in the cost function. So, the fixed costs are \(\boxed{1800}\) dollars.
Step 1 ：First, we need to find the average cost function. We know that the derivative of the average cost function is given by \(\bar{C}^\prime(x) = -\frac{1800}{x^2}\). So, the average cost function \(\bar{C}(x)\) is the antiderivative of \(\bar{C}^\prime(x)\).
Step 2 ：We can find the antiderivative of \(\bar{C}^\prime(x)\) by integrating \(\bar{C}^\prime(x)\) with respect to \(x\).
Step 3 ：\[\int \bar{C}^\prime(x) dx = \int -\frac{1800}{x^2} dx = -1800 \int \frac{1}{x^2} dx\]
Step 4 ：Using the power rule for integration, we get \(-1800 \int x^{-2} dx = -1800(-x^{-1}) + C = 1800\frac{1}{x} + C\), where \(C\) is the constant of integration.
Step 5 ：We know that \(\bar{C}(100) = 21\), so we can substitute \(x = 100\) into the equation to solve for \(C\).
Step 6 ：\[21 = 1800\frac{1}{100} + C\Rightarrow C = 21 - 18 = 3\]
Step 7 ：So, the average cost function is \(\bar{C}(x) = 1800\frac{1}{x} + 3\).
Step 8 ：Next, we need to find the cost function. The cost function \(C(x)\) is the product of the average cost function \(\bar{C}(x)\) and \(x\).
Step 9 ：So, \(C(x) = x\bar{C}(x) = x(1800\frac{1}{x} + 3) = 1800 + 3x\).
Step 10 ：The fixed costs are the costs that do not change with the quantity of output, which is the constant term in the cost function. So, the fixed costs are \(\boxed{1800}\) dollars.
