Question

The Art of Problem Solving has begun selling a cookbook called "What Would Euler Eat?" If the price of the cookbook is $n$ dollars ($n \le 60$), then it will sell $300n - 5n^2$ copies. What price per book (in dollars) will maximize the total revenue we receive for all the books sold?

Answer 1

The total revenue is calculated by multiplying the price per book by the number of copies sold. In this case, the revenue function \(R\) is given by the product of the price \(n\) and the expression for the number of copies sold \(300n - 5n^2\):

\[ R(n) = n \cdot (300n - 5n^2) \]

To find the price per book that maximizes revenue, you need to find the critical points of this function. Take the derivative of \(R(n)\) with respect to \(n\), set it equal to zero, and solve for \(n\). Once you find the value of \(n\), substitute it back into the \(R(n)\) equation to get the maximum revenue.

After finding the critical points, check the endpoints of the given interval \((n \le 60)\) to ensure you've considered all possible cases.