John owns a hotdog stand. He has found that his profit is represented by the equation $P(x)=-x^{2}+78 x-80$, with $P$ being profits, in cents, and $x$ the number of hotdogs sold. How many hotdogs must he sell to earn the most profit? A. 40 hotdogs B. 41 hotdogs C. 21 hotdogs D. 39 hotdogs

Step 1 ：John owns a hotdog stand. He has found that his profit is represented by the equation \(P(x)=-x^{2}+78 x-80\), with \(P\) being profits, in cents, and \(x\) the number of hotdogs sold. How many hotdogs must he sell to earn the most profit?

Step 2 ：The profit function is a quadratic function. The maximum or minimum of a quadratic function is at its vertex. The x-coordinate of the vertex of a quadratic function given in the form \(f(x) = ax^2 + bx + c\) is \(-b/2a\).

Step 3 ：In this case, \(a = -1\) and \(b = 78\).

Step 4 ：So, the number of hotdogs he must sell to earn the most profit is \(-b/2a = -78/2*(-1) = 39\).

Step 5 ：Final Answer: \(\boxed{39 \text{ hotdogs}}\).