Final answer:
The equation to model the soccer ball's path is v = -2/135(x - 45)^2 + 30, where a = -2/135, h = 45 feet, and k = 30 feet.
Step-by-step explanation:
To find the values of a, h, and k in the vertex form equation v = a(x-h)^2 + k that models the parabolic path of a soccer ball kicked from the origin, we need to use the given information: the ball travels a horizontal distance of 90 feet and reaches a maximum height of 30 feet.
Since the path of the ball is parabolic and symmetric, the maximum height will occur at the midpoint of the horizontal distance. Therefore, h = 45 feet, which is half of 90 feet. The value of k is the maximum height, which is given as 30 feet.
Now we can determine the value of a. We know that the parabola passes through the origin, so we can substitute (0,0) for (x,v) along with our values for h and k. Thus, we get:
0 = a(0 - 45)^2 + 30
To solve for a, we first isolate the a on one side:
a = -30/45^2
This simplifies to:
a = -30/2025
a = -2/135 (leaving as an unreduced fraction as instructed)
Therefore, the equation to model the path of the soccer ball is:
v = -2/135(x - 45)^2 + 30