The motion of the golf ball can be modeled by a parabolic equation. Since Lorraine hits the ball and it forms a parabolic trajectory, we can use the standard form of a parabola given by:
\[ h(x) = a(x - h)^2 + k \]
where:
- \(h(x)\) is the height of the ball at distance \(x\),
- \((h, k)\) is the vertex of the parabola.
In this case, the vertex of the parabola occurs at the highest point of the ball. We know:
- The highest point (vertex) is at \( (30, 30) \) (30 feet in the air and the distance from Lorraine),
- The ball lands at \( (60, 0) \).
We can find the value of \(a\) by using the point where the ball touches the ground, which is at \(x = 60\) and \(h(60) = 0\):
Using the vertex form of the equation:
\[ h(x) = a(x - 30)^2 + 30 \]
Substituting the point (60, 0):
\[ 0 = a(60 - 30)^2 + 30 \]
This simplifies to:
\[ 0 = a(30^2) + 30 \] \[ 0 = 900a + 30 \] \[ 900a = -30 \] \[ a = -\frac{30}{900} = -\frac{1}{30} \]
Now we have the equation of the height of the ball:
\[ h(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Now we will evaluate the function to find the height when the ball is 15 feet away from Lorraine (i.e., at \(x = 15\)):
\[ h(15) = -\frac{1}{30}(15 - 30)^2 + 30 \] \[ = -\frac{1}{30}(-15)^2 + 30 \] \[ = -\frac{1}{30}(225) + 30 \] \[ = -\frac{225}{30} + 30 \] \[ = -7.5 + 30 \] \[ = 22.5 \]
Thus, the height of the golf ball when it is 15 feet away from Lorraine is 22.5 feet.
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