John owns a hot dog stand. He has found that his profit is given by the equation $P = - x ^ { 2 } + 80 x + 87$ , where x is the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?
A) 23 hot dogs
B) 47 hot dogs
C) 40 hot dogs
D) 41 hot dogs

Question

Quizplus · Accepted Answer

The profit function $P = -x^2 + 80x + 87$ is a quadratic equation that opens downwards (since the coefficient of $x^2$ is negative). The maximum profit occurs at the vertex of the parabola. The x-coordinate of the vertex is given by $-b/(2a)$, where $a = -1$ and $b = 80$. Calculating this gives $x = 40$.

John Owns a Hot Dog Stand P=−x2+80x+87P = - x ^ { 2 } + 80 x + 87P=−x2+80x+87

John Owns a Hot Dog Stand $P = - x ^ { 2 } + 80 x + 87$