Chapter 6.3, Problem 22E

In Exercises 15-22, assume that relative maximum and minimum values are absolute maximum and minimum values.

Maximizing profit. McLeod Corp. finds that its profit, P, in millions of dollars, is given by

$P\left(a,p\right)=2ap+80p-15p^2-\frac{1}{10}{a}^{2}p-80,$ where a is the amount spent on advertising, in millions of dollars, and p is the price charged per unit, in dollars. Find the maximum value of P and the values of a and p at which it occurs.