John owns a hot dog stand. He has found that his profit is given by the equation $\mathrm{P}(\mathrm{x})=-\mathrm{x}^{2}+70 \mathrm{x}+82$, where $\mathrm{x}$ is the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?

Question

Quizplus · Accepted Answer

The profit function $P(x) = -x^2 + 70x + 82$ 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 gives the maximum profit. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$. Here, $a = -1$ and $b = 70$, so $-\frac{70}{2(-1)} = 35$. Therefore, John must sell 35 hot dogs to earn the most profit.

John Owns a Hot Dog Stand P(x)=−x2+70x+82\mathrm{P}(\mathrm{x})=-\mathrm{x}^{2}+70 \mathrm{x}+82P(x)=−x2+70x+82 , Where x\mathrm{x}x Is the Number of Hot Dogs Sold

John Owns a Hot Dog Stand $\mathrm{P}(\mathrm{x})=-\mathrm{x}^{2}+70 \mathrm{x}+82$ , Where $\mathrm{x}$ Is the Number of Hot Dogs Sold