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?(1 point) feet
Answers
GPT-4o mini
To find the parabolic equation that describes the height of the golf ball as it moves away from Lorraine, we can model the height \( h(x) \) of the ball as a quadratic function.
Given:
- The maximum height of the ball is 30 feet, and it occurs at the vertex of the parabola.
- The ball lands 60 feet away from Lorraine, which gives the roots of the parabola at \( x = 0 \) (where Lorraine is) and \( x = 60 \).
We can use the vertex form of a quadratic equation, which is:
\[
h(x) = a(x - h)^2 + k
\]
where \((h, k)\) are the coordinates of the vertex. In our case, the vertex is at \((30, 30)\) since the highest point is at 30 feet.
1. The vertex is at \((30, 30)\).
2. The roots (where the height is zero) are at \( x = 0 \) and \( x = 60 \).
### Step-by-step derivation:
1. Since the parabola opens downwards (as it reaches a maximum height), we'll express the equation as:
\[
h(x) = a(x - 30)^2 + 30
\]
2. To find the value of \( a \):
- We know \( h(0) = 0 \) (height when the ball is at Lorraine).
Substituting \( x = 0 \):
\[
0 = a(0 - 30)^2 + 30
\]
\[
0 = a(900) + 30
\]
\[
a(900) = -30
\]
\[
a = -\frac{30}{900} = -\frac{1}{30}
\]
3. Thus, the height function can be written as:
\[
h(x) = -\frac{1}{30}(x - 30)^2 + 30
\]
### Determine the height when \( x = 15 \) feet:
Now we need to evaluate \( h(15) \):
\[
h(15) = -\frac{1}{30}(15 - 30)^2 + 30
\]
\[
= -\frac{1}{30}(-15)^2 + 30
\]
\[
= -\frac{1}{30}(225) + 30
\]
\[
= -\frac{225}{30} + 30
\]
\[
= -7.5 + 30
\]
\[
= 22.5
\]
Therefore, the height of the golf ball when it is 15 feet away from Lorraine is **22.5 feet**.
Given:
- The maximum height of the ball is 30 feet, and it occurs at the vertex of the parabola.
- The ball lands 60 feet away from Lorraine, which gives the roots of the parabola at \( x = 0 \) (where Lorraine is) and \( x = 60 \).
We can use the vertex form of a quadratic equation, which is:
\[
h(x) = a(x - h)^2 + k
\]
where \((h, k)\) are the coordinates of the vertex. In our case, the vertex is at \((30, 30)\) since the highest point is at 30 feet.
1. The vertex is at \((30, 30)\).
2. The roots (where the height is zero) are at \( x = 0 \) and \( x = 60 \).
### Step-by-step derivation:
1. Since the parabola opens downwards (as it reaches a maximum height), we'll express the equation as:
\[
h(x) = a(x - 30)^2 + 30
\]
2. To find the value of \( a \):
- We know \( h(0) = 0 \) (height when the ball is at Lorraine).
Substituting \( x = 0 \):
\[
0 = a(0 - 30)^2 + 30
\]
\[
0 = a(900) + 30
\]
\[
a(900) = -30
\]
\[
a = -\frac{30}{900} = -\frac{1}{30}
\]
3. Thus, the height function can be written as:
\[
h(x) = -\frac{1}{30}(x - 30)^2 + 30
\]
### Determine the height when \( x = 15 \) feet:
Now we need to evaluate \( h(15) \):
\[
h(15) = -\frac{1}{30}(15 - 30)^2 + 30
\]
\[
= -\frac{1}{30}(-15)^2 + 30
\]
\[
= -\frac{1}{30}(225) + 30
\]
\[
= -\frac{225}{30} + 30
\]
\[
= -7.5 + 30
\]
\[
= 22.5
\]
Therefore, the height of the golf ball when it is 15 feet away from Lorraine is **22.5 feet**.