John owns a hot dog stand. He has found that his profit is represented by the equation p(x)=-x^2+62x+77, with p being profits and x the number of hot dogs sold. How many hot dogs must he sell to earn the most profit?

We want to maximize the profit, p(x) = -x^2 + 62x + 77.  
The graph of this function is a parabola that opens down.  The x-coordinate of the vertex is x = -b/(2a), which here is x = -62 / (2*-1), or 31.  So, we'll maximize John's profit if we advise him to sell 31 hot dogs.