John owns a hot dog stand. He has found that his profit is given by the equation $P=-x^{2}+50 x+71$, where $\mathrm{x}$ is the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?
A) 46 hot dogs
B) 26 hot dogs
C) 25 hot dogs
D) 23 hot dogs

Question

Quizplus · Accepted Answer

The profit equation $P=-x^{2}+50x+71$ 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 $-\frac{b}{2a}$, represents the number of hot dogs sold for maximum profit. Here, $a = -1$, $b = 50$, so the vertex is at $x = -\frac{50}{2(-1)} = 25$. Therefore, John must sell 25 hot dogs to earn the most profit.

John Owns a Hot Dog Stand P=−x2+50x+71P=-x^{2}+50 x+71P=−x2+50x+71 , Where x\mathrm{x}x Is the Number of Hot Dogs Sold

John Owns a Hot Dog Stand $P=-x^{2}+50 x+71$ , Where $\mathrm{x}$ Is the Number of Hot Dogs Sold