A car rental agency rents 480 cars per day at a rate of $\$ 30$ per day. For each $\$ 1$ increase in rate, 10 fewer cars are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum income?
The rental agency will earn a maximum income of $\$$ when it charges $\$$ per day.
Answer
Final Answer: The cars should be rented at a rate of \(\boxed{24}\) dollars per day to produce the maximum income. The maximum income is \(\boxed{11520}\) dollars.
Step 1 ：Let's denote the rate as 'r'. The number of cars rented can be expressed as a function of the rate: \(c = 480 - 10r\).
Step 2 ：The income of the agency is the product of the number of cars rented and the rate per day, which can be expressed as \(I = r * (480 - 10r)\).
Step 3 ：To find the maximum income, we need to find the derivative of the income function with respect to the rate, set it equal to zero, and solve for 'r'. The derivative of the income function is \(I' = 480 - 20r\).
Step 4 ：Setting the derivative equal to zero gives us \(480 - 20r = 0\). Solving for 'r' gives us \(r = 24\).
Step 5 ：Substituting \(r = 24\) back into the income function gives us the maximum income: \(I = 24 * (480 - 10*24) = 11520\).
Step 6 ：Final Answer: The cars should be rented at a rate of \(\boxed{24}\) dollars per day to produce the maximum income. The maximum income is \(\boxed{11520}\) dollars.
