College Algebra, Chapter 4, 4.1, Section 4.1, Problem 66

Suppose a soft drink vendor at a popular beach analyzes his sales records and finds that if he sells $x$ of cans of soda pop in one day, his profit (in dollars) is given by

$P(x) = -0.001x^2 + 3x - 1800$

What is his maximum profit per day, and how many cans must he sell for maximum profit?

The function $P$ is a quadratic function with $a = -0.001$ and $b = 3$. Thus, its maximum value occurs when

$\displaystyle x = - \frac{b}{2a} = - \frac{3}{2(-0.001)} = 1500$ cans

The maximum profit is $P(1,500) = -0.001 (1,500)^2 + 3(1,500) - 1800 = \$ 450$.

So the vendor needs to sell $1500$ pieces of cans of soda pop in order to have a maximum profit of $\$ 450$.