A bookstore is deciding what price it should charge for a certain book. After research, the store finds that if the book's price is $p$ dollars (where $p \le 26$), then the number of books sold per month is $130-5p$. What price should the store charge to maximize its revenue?
I want to find the value of $p$ that maximizes the revenue, which is the product of the price and the number of books sold. So I need to write an expression for the revenue in terms of $p$ and then find its maximum value. The expression for the revenue is $R(p) = p(130-5p)$. To find the maximum value, I can use calculus and find the critical points where the derivative is zero or undefined. The derivative of $R(p)$ is $R'(p) = 130-10p$. This is defined for all values of $p$, so I only need to set it equal to zero and solve for $p$. $R'(p) = 0$ when $p = 13$. To check if this is a maximum or a minimum, I can use the second derivative test or look at the graph of $R(p)$. The second derivative of $R(p)$ is $R''(p) = -10$, which is negative for all values of $p$. This means that $R(p)$ is concave down and $p = 13$ is a maximum point. Either way, I conclude that the store should charge $13$ dollars for the book to maximize its revenue. 