What is the maximum number of oranges that can be harvested according to the function h(x) = -10x² + 280x + 17,400?

To determine the maximum number of oranges that can be harvested according to the quadratic function h(x) = -10x² + 280x + 17,400, we need to find the vertex of the parabola represented by this function. Since the coefficient of x² is negative, the parabola opens downwards, indicating that the vertex will indeed give us the maximum value of the function.

The x-coordinate of the vertex of a quadratic function in the form ax² + bx + c is given by the formula:

[ x = -\frac{b}{2a} ]

In this case, a is -10 and b is 280. Plugging in these values:

[ x = -\frac{280}{2 \times -10} ]

[ x = -\frac{280}{-20} ]

[ x = 14 ]

Now, we substitute this value back into the original function to find h(14), which will give us the maximum number of oranges:

[ h(14) = -10(14)² + 280(14) + 17,400 ]

Calculating each term:

1. First calculate ( 14² ):

[ 14²