John owns a hot dog stand. He has found that his profit is represented by the equation $P ( x ) = - x ^ { 2 } + 56 x + 72$ with P being profits and x the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?
A) 28 hot dogs
B) 29 hot dogs
C) 44 hot dogs
D) 22 hot dogs

Question

Quizplus · Accepted Answer

The profit equation $P(x) = -x^2 + 56x + 72$ is a quadratic equation in the form of $ax^2 + bx + c$, where the coefficient of $x^2$ is negative, indicating a downward-opening parabola. The vertex of this parabola, given by the formula $x = -\frac{b}{2a}$, represents the point of maximum profit. Here, $a = -1$ and $b = 56$, so $x = -\frac{56}{2(-1)} = 28$. Therefore, John must sell 28 hot dogs to earn the most profit.

John Owns a Hot Dog Stand P(x)=−x2+56x+72P ( x ) = - x ^ { 2 } + 56 x + 72P(x)=−x2+56x+72

John Owns a Hot Dog Stand $P ( x ) = - x ^ { 2 } + 56 x + 72$