Solve.
-John owns a hotdog stand. His profit is represented by $P ( x ) = - x ^ { 2 } + 14 x + 55$, with $P ( x )$ being profit and $x$ the number of hotdogs sold. What is the most he can earn?
A) $\$ 49$
B) $\$ 104$
C) $\$ 111$
D) $\$ 90$

Question

Quizplus · Accepted Answer

The given profit function $P(x) = -x^2 + 14x + 55$ is a quadratic equation in the form of $ax^2 + bx + c$, where $a = -1$, $b = 14$, and $c = 55$. The vertex form of a quadratic function $y = ax^2 + bx + c$ is given by $y = a(x - h)^2 + k$, where the vertex $(h, k)$ represents the maximum or minimum point of the function, depending on the sign of $a$. Since $a = -1$ is negative, the parabola opens downwards, and the vertex represents the maximum point of the function. The x-coordinate of the vertex can be found using the formula $h = -\frac{b}{2a}$. Substituting $a = -1$ and $b = 14$ into the formula gives $h = -\frac{14}{2(-1)} = 7$. To find the maximum profit ($k$), substitute $x = 7$ into the original profit function: $P(7) = -(7)^2 + 14(7) + 55 = -49 + 98 + 55 = 104$. Therefore, the maximum profit John can earn is $104.

Solve -John Owns a Hotdog Stand P(x)=−x2+14x+55P ( x ) = - x ^ { 2 } + 14 x + 55P(x)=−x2+14x+55

Solve
-John Owns a Hotdog Stand $P ( x ) = - x ^ { 2 } + 14 x + 55$