John owns a hotdog stand. His profit is represented by the equation $P(x)=-x^{2}+12 x+41$, with $P$ being profits and $x$ the number of hotdogs sold. What is the most he can earn? A. $\$ 36$ B. $\$ 83$ C. $\$ 77$ D. $\$ 65$

Step 1 ：John owns a hotdog stand. His profit is represented by the equation \(P(x)=-x^{2}+12 x+41\), with \(P\) being profits and \(x\) the number of hotdogs sold. What is the most he can earn?

Step 2 ：The profit function is a quadratic function. The maximum value of a quadratic function \(f(x) = ax^2 + bx + c\) where \(a < 0\) is at the vertex of the parabola.

Step 3 ：The x-coordinate of the vertex can be found using the formula \(-\frac{b}{2a}\). In this case, \(a = -1\) and \(b = 12\). So, we can substitute these values into the formula to find the x-coordinate of the vertex.

Step 4 ：Then, we substitute this x-coordinate into the profit function to find the maximum profit.

Step 5 ：The correct x-coordinate of the vertex is \(\frac{-12}{2*-1} = 6\).

Step 6 ：We substitute this correct x-coordinate into the profit function to find the maximum profit.

Step 7 ：Final Answer: The most John can earn is \(\boxed{\$ 77}\).