Suppose that the marginal revenue for firefighting protective clothes is $\overline{M R}=263-2 x$ and the marginal cost is $\overline{M C}=2.5 x+11$ with a fixed cost of $\$ 270$. Assume $R(0)=0$. b) Find the revenue function. \[ R(x)= \] c) Find the cost function. \[ C(x)= \]
Solution
Step 1 ：Given that the marginal revenue for firefighting protective clothes is \(\overline{M R}=263-2 x\) and the marginal cost is \(\overline{M C}=2.5 x+11\) with a fixed cost of $270. Assume \(R(0)=0\).
Step 2 ：We need to find the revenue function and the cost function. The revenue function \(R(x)\) is the integral of the marginal revenue function, and the cost function \(C(x)\) is the integral of the marginal cost function plus the fixed cost.
Step 3 ：First, we find the revenue function by integrating the marginal revenue function. The integral of \(263 - 2x\) is \(-x^2 + 263x\). So, the revenue function is \(R(x) = -x^2 + 263x\).
Step 4 ：Next, we find the cost function by integrating the marginal cost function and adding the fixed cost. The integral of \(2.5x + 11\) is \(1.25x^2 + 11x\). Adding the fixed cost of $270, the cost function is \(C(x) = 1.25x^2 + 11x + 270\).
Step 5 ：\(\boxed{\text{Final Answer: The revenue function is } R(x) = -x^2 + 263x \text{ and the cost function is } C(x) = 1.25x^2 + 11x + 270}\)
