Question

Lorraine hits a golf ball into the air. It lands 60 feet away from her. At its highest point, the ball is 30 feet in the air. Find the parabolic equation that describes the height of the ball as it moves away from Lorraine. Use this equation to evaluate the function to answer the following question: What is the height of the golf ball when it is 15 feet away from Lorraine?(5 points)

Answer 1

To find the parabolic equation that describes the height of the golf ball as it moves away from Lorraine, we will assume the vertex form of a parabola, which is given by:

\[
h(x) = a(x - h)^2 + k
\]

Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) is a constant that determines the width and direction of the parabola.

In this scenario:
- The vertex of the parabola, where the ball reaches its maximum height, is at \( (30, 30) \) (30 feet in the air).
- The ball lands 60 feet away from Lorraine at a height of 0. Thus, we have another point \( (60, 0) \).

Substituting these points into the vertex form, we now have two equations:

1. Vertex (30, 30):
\[
h(x) = a(x - 30)^2 + 30
\]

2. The point (60, 0):
\[
0 = a(60 - 30)^2 + 30
\]
Simplifying, we get:
\[
0 = a(30)^2 + 30
\]

This can be rewritten to solve for \( a \):
\[
0 = 900a + 30
\]
\[
900a = -30
\]
\[
a = -\frac{1}{30}
\]

Now we can substitute \( a \) back into the vertex form:
\[
h(x) = -\frac{1}{30}(x - 30)^2 + 30
\]

This is the parabolic equation of the height of the ball.

Next, we want to find the height of the golf ball when it is 15 feet away from Lorraine, which means we need to find \( h(15) \).

Substituting \( x = 15 \) into the equation:
\[
h(15) = -\frac{1}{30}(15 - 30)^2 + 30
\]
\[
= -\frac{1}{30}(-15)^2 + 30
\]
\[
= -\frac{1}{30}(225) + 30
\]
\[
= -7.5 + 30
\]
\[
= 22.5
\]

Thus, the height of the golf ball when it is 15 feet away from Lorraine is **22.5 feet**.