Question

The path of water projected from a fountain can be modeled by the equation y=-15x^2+120x, where x is the horizontal distance from the nozzle and y is the height. Find the average rate of change of the waters height over the interval [0,4]

Answer 1

The average rate of change of the water's height over the interval [0,4] is 60 meters per horizontal meter, calculated using the difference in function values over the interval divided by the interval's length.

Step-by-step explanation:

The student is asking about the average rate of change of a quadratic function, which represents the path of water projected from a fountain. To find the average rate of change over the interval [0,4], we apply the formula for average rate of change:

average rate of change = (f(b) - f(a)) / (b - a)

Using the given equation y=-15x^2+120x, we calculate y at x=0 and x=4:

• f(0) = -15(0)^2 + 120(0) = 0
• f(4) = -15(4)^2 + 120(4) = -15(16) + 480 = -240 + 480 = 240

Now, substituting these values into the average rate of change formula, we get:

(240 - 0) / (4 - 0) = 240 / 4 = 60

Therefore, the average rate of change of the water's height over the interval [0,4] is 60 meters per horizontal meter.