Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, $R(x)$, and cost, $C(x)$, of producing $x$ units are in dollars. \[ R(x)=50 x-0.1 x^{2}, C(x)=5 x+30 \]
Solution
Step 1 ：Given the revenue function, \(R(x) = 50x - 0.1x^2\), and the cost function, \(C(x) = 5x + 30\), we can find the profit function, \(P(x)\), by subtracting the cost function from the revenue function. This gives us \(P(x) = -0.1x^2 + 45x - 30\).
Step 2 ：To find the maximum profit, we need to find the maximum point of the profit function. This can be done by finding the derivative of the profit function and setting it equal to zero to find the critical points. The derivative of the profit function, \(P'(x)\), is \(45 - 0.2x\).
Step 3 ：Setting \(P'(x)\) equal to zero gives us the equation \(45 - 0.2x = 0\). Solving for \(x\) gives us the critical point \(x = 225\).
Step 4 ：Substituting \(x = 225\) into the profit function, \(P(x)\), gives us the maximum profit, \(P(225) = 5032.50\).
Step 5 ：Final Answer: The maximum profit is \(\boxed{5032.50}\) and the number of units that must be produced and sold in order to yield the maximum profit is \(\boxed{225}\) units.
