John owns a hot dog stand. His profit, in dollars, is given by the function $P(x)=-x^{2}+12 x+44$, where $\mathrm{x}$ is the number of hot dogs sold. What is the most he can earn?

Question

Quizplus · Accepted Answer

The profit function $P(x)=-x^{2}+12x+44$ is a quadratic equation that opens downwards (since the coefficient of $x^2$ is negative), indicating that its vertex represents the maximum profit. The x-coordinate of the vertex of a parabola given by $ax^2 + bx + c$ is $-\frac{b}{2a}$. For this function, $a = -1$ and $b = 12$, so the x-coordinate is $-\frac{12}{2(-1)} = 6$. Plugging this back into the profit function gives $P(6) = -(6)^2 + 12(6) + 44 = -36 + 72 + 44 = 80$, which means the maximum profit John can earn is $80.

John Owns a Hot Dog Stand P(x)=−x2+12x+44P(x)=-x^{2}+12 x+44P(x)=−x2+12x+44 , Where x\mathrm{x}x Is the Number of Hot Dogs Sold

John Owns a Hot Dog Stand $P(x)=-x^{2}+12 x+44$ , Where $\mathrm{x}$ Is the Number of Hot Dogs Sold