(1 point) The total cost (in hundreds of dollars) of producing \( x \) slide rules per day is \[ C(x)=8+\sqrt{2 x+18} \] Find the marginal cost at each production level \( x \) given below: \[ \begin{array}{l} x=25 \\ x=43 \end{array} \]

To find the marginal cost, we first need to compute the derivative of the total cost function \( C(x) \). The total cost function is given by: \[ C(x) = 8 + \sqrt{2x + 18} \] The marginal cost \( C'(x) \) is the derivative of \( C(x) \). Let's differentiate: 1. The derivative of the constant \( 8 \) is \( 0 \). 2. For the term \( \sqrt{2x + 18} \), we use the chain rule. The derivative is: \[ C'(x) = \frac{1}{2\sqrt{2x + 18}} \cdot (2) = \frac{1}{\sqrt{2x + 18}} \] Now we can compute the marginal cost at the specified production levels. **At \( x = 25 \):** \[ C'(25) = \frac{1}{\sqrt{2(25) + 18}} = \frac{1}{\sqrt{50 + 18}} = \frac{1}{\sqrt{68}} = \frac{1}{\sqrt{4 \cdot 17}} = \frac{1}{2\sqrt{17}} \] Calculating \( \sqrt{17} \approx 4.123 \), we get: \[ C'(25) \approx \frac{1}{2 \cdot 4.123} \approx \frac{1}{8.246} \approx 0.121 \] Since the cost function is in hundreds of dollars, the marginal cost at \( x = 25 \) is approximately \( 0.121 \times 100 \approx 12.1 \) dollars. **At \( x = 43 \):** \[ C'(43) = \frac{1}{\sqrt{2(43) + 18}} = \frac{1}{\sqrt{86 + 18}} = \frac{1}{\sqrt{104}} = \frac{1}{\sqrt{4 \cdot 26}} = \frac{1}{2\sqrt{26}} \] Calculating \( \sqrt{26} \approx 5.099 \), we get: \[ C'(43) \approx \frac{1}{2 \cdot 5.099} \approx \frac{1}{10.198} \approx 0.098 \] Again, converting to dollars, the marginal cost at \( x = 43 \) is approximately \( 0.098 \times 100 \approx 9.8 \) dollars. In summary: - The marginal cost at \( x = 25 \) is approximately **$12.10**. - The marginal cost at \( x = 43 \) is approximately **$9.80**.