To find the number of hot dogs that must be sold to earn the maximum profit, we need to determine the vertex of the parabolic function represented by the given equation P = x² + 70x + 83. The vertex of a parabola corresponds to the maximum or minimum point of the function.
1. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are coefficients.
2. In this case, the given equation P = x² + 70x + 83 matches the general form with a = 1, b = 70, and c = 83.
3. To find the x-coordinate of the vertex, we can use the formula x = -b/(2a) for the vertex of a quadratic function in the form f(x) = ax² + bx + c.
4. Substituting the values a = 1 and b = 70 into the formula x = -b/(2a), we get x = -70/(2*1) = -35.
5. Therefore, the number of hot dogs that must be sold to earn the maximum profit is 35.
By selling 35 hot dogs, the profit will reach its maximum value based on the given quadratic function.
