To model the height of the golf ball as a function of its distance from Lorraine, we can use the standard form of a parabolic equation:
\[ h(x) = -a(x - h)^2 + k \]
where:
- \( (h, k) \) is the vertex of the parabola,
- \( x \) is the distance from Lorraine,
- \( a \) is a coefficient that determines the width and direction of the parabola.
Step 1: Identify the vertex
- The vertex of the parabola occurs at the highest point of the golf ball. Given that the highest point is 30 feet and it is at the midpoint of the distance traveled, the vertex is at \( (30, 30) \).
Step 2: Identify the x-intercepts
- The ball lands 60 feet away, which means the ball starts from \( (0, 0) \) and lands at \( (60, 0) \).
- Thus, the x-intercepts are at \( (0, 0) \) and \( (60, 0) \).
Step 3: Construct the equation
The equation in vertex form is:
\[ h(x) = -a(x - 30)^2 + 30 \]
We need to compute the value of \( a \) using one of the x-intercepts. Let's use \( (0, 0) \).
- Substitute \( (0, 0) \) into the equation:
\[ 0 = -a(0 - 30)^2 + 30 \]
This simplifies to:
\[ 0 = -a(900) + 30 \]
\[ a(900) = 30 \]
\[ a = \frac{30}{900} = \frac{1}{30} \]
Step 4: Write the complete equation
Now we can substitute \( a \) back into the vertex form:
\[ h(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Step 5: Find the height when the ball is 15 feet away
Next, we evaluate \( h(15) \):
\[ h(15) = -\frac{1}{30}(15 - 30)^2 + 30 \]
Calculating this gives:
\[ h(15) = -\frac{1}{30}(-15)^2 + 30 \]
\[ h(15) = -\frac{1}{30}(225) + 30 \]
\[ h(15) = -\frac{225}{30} + 30 \]
\[ h(15) = -7.5 + 30 \]
\[ h(15) = 22.5 \]
Conclusion
Thus, the height of the golf ball when it is 15 feet away from Lorraine is \( \boxed{22.5} \) feet.
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