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.

Using a graphing calculator you can find that the maximum is 1038, so the profit starts to decline at the t value for 1038, which is 31