The weekly marginal cost of producing $x$ pairs of tennis shoes is given by the following function, where $C(x)$ is $\operatorname{cost}$ in dollars. \[ C^{\prime}(x)=13+\frac{200}{x+1} \] If the fixed costs are $\$ 4,000$ per week, find the cost function. What is the average cost per pair of shoes if 1,000 pairs of shoes are produced each week?
Solution
Step 1 ：The weekly marginal cost of producing \(x\) pairs of tennis shoes is given by the function \(C^\prime(x) = 13 + \frac{200}{x+1}\), where \(C(x)\) is the cost in dollars.
Step 2 ：The cost function can be found by integrating the marginal cost function. The integral of \(C^\prime(x)\) is \(\int C^\prime(x) dx = \int (13 + \frac{200}{x+1}) dx = 13x + 200\log(x + 1)\).
Step 3 ：Adding the fixed costs of $4000 per week to the cost function gives us the total cost function \(C(x) = 13x + 200\log(x + 1) + 4000\).
Step 4 ：The average cost per pair of shoes can be found by dividing the total cost by the number of pairs produced. If 1000 pairs of shoes are produced each week, the average cost is \(\frac{C(1000)}{1000} = \frac{13*1000 + 200\log(1001) + 4000}{1000} = \log(1001)/5 + 17\).
Step 5 ：\(\boxed{\text{The cost function is } C(x) = 13x + 200\log(x + 1) + 4000 \text{ and the average cost per pair of shoes when 1000 pairs of shoes are produced each week is } \log(1001)/5 + 17}\)
