Question

Tavon wants to design a fountain for his backyard. He wants the height of the fountain's water stream to be at least 2 m. Tavon wants the fountain's water to stream in four directions. He puts four nozzles in the center of the pool so there will be four streams originating from the fountain. Assume that the quadratic function h(x) = -12x² + 12x models the path of a jet of water with a constant pressure, where x is the horizontal distance from the jet of water to the nozzle and h is the height of the jet of water. What is the maximum height of the jet of water?

1) 2 m
2) 4 m
3) 6 m
4) 8 m

Answer 1

The maximum height of the jet of water from Tavon's fountain, modeled by the quadratic function h(x) = -12x² + 12x, is calculated to be 3 meters.

Step-by-step explanation:

To find the maximum height that the jet of water reaches in Tavon's fountain design, we analyze the given quadratic function h(x) = -12x² + 12x. This function is in the form -ax² + bx, which is a downward-opening parabola, indicating that the maximum height occurs at the vertex of the parabola. The x-coordinate of the vertex of a parabola in this form is given by -b/(2a). Substituting the values from our equation gives us:

x = -12 / (2 * -12) = 1/2

Now, we plug this value back into the quadratic function to find the maximum height (h):

h(1/2) = -12*(1/2)² + 12*(1/2) = -12*1/4 + 12*1/2 = -3 + 6 = 3 m

Therefore, the maximum height of the jet of water is 3 meters.