John owns a hot dog stand. His profit, in dollars, is given by the equation $P ( x ) = P = - x ^ { 2 } + 14 x + 59$ , where x is the number of hot dogs sold. What is the most he can earn?
A) $150
B) $108
C) $49
D) $87

Question

Quizplus · Accepted Answer

The profit function $P(x) = -x^2 + 14x + 59$ 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 $b = 14$. Calculating this gives $x = 7$. Substituting $x = 7$ into the profit equation gives $P(7) = -(7)^2 + 14(7) + 59 = 108$. Thus, the maximum profit is $108.

John Owns a Hot Dog Stand P(x)=P=−x2+14x+59P ( x ) = P = - x ^ { 2 } + 14 x + 59P(x)=P=−x2+14x+59

John Owns a Hot Dog Stand $P ( x ) = P = - x ^ { 2 } + 14 x + 59$