Solve.
-John owns a hotdog stand. He has found that his profit is represented by the function $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 74 \mathrm { x } + 80$, with $\mathrm { P } ( \mathrm { x } )$ being profits and $x$ the number of hotdogs sold. How many hotdogs must he sell to earn the most profit?
A) 21 hotdogs
B) 43 hotdogs
C) 38 hotdogs
D) 37 hotdogs

Question

Quizplus · Accepted Answer

The profit function $P(x) = -x^2 + 74x + 80$ 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 number of hotdogs sold for maximum profit. Here, $a = -1$, $b = 74$, so $x = -\frac{74}{2(-1)} = 37$. Therefore, John must sell 37 hotdogs to earn the most profit.

Solve -John Owns a Hotdog Stand P(x)=−x2+74x+80\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 74 \mathrm { x } + 80P(x)=−x2+74x+80

Solve
-John Owns a Hotdog Stand $\mathrm { P } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 74 \mathrm { x } + 80$