To find the parabolic equation that describes the height of the golf ball as it moves away from Lorraine, we can use the vertex form of a quadratic equation:
\[ h(x) = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex, and \( a \) is a constant that determines the width and direction of the parabola.
From the problem, we know:
- The ball's maximum height (30 feet) occurs at the vertex of the parabola.
- The maximum height is at the midpoint of the distance it travels horizontally, which is \( x = 30 \) feet (half of 60 feet).
- The vertex (highest point) is therefore \( (30, 30) \).
Now we can write the vertex form of the equation as:
\[ h(x) = a(x - 30)^2 + 30 \]
Next, we need to determine the value of \( a \). We know that when the ball is 60 feet away (\( x = 60 \)), the height \( h \) is 0 (the ball lands). We can substitute these values into our equation to solve for \( a \):
\[ 0 = a(60 - 30)^2 + 30 \]
Calculating:
\[ 0 = a(30)^2 + 30 \] \[ 0 = 900a + 30 \] \[ 900a = -30 \] \[ a = -\frac{1}{30} \]
Now we have the complete equation for the height of the ball:
\[ h(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Next, we want to find the height of the golf ball when it is 15 feet away from Lorraine (\( x = 15 \)):
\[ h(15) = -\frac{1}{30}(15 - 30)^2 + 30 \]
Calculating:
\[ h(15) = -\frac{1}{30}(-15)^2 + 30 \] \[ h(15) = -\frac{1}{30}(225) + 30 \] \[ h(15) = -7.5 + 30 \] \[ h(15) = 22.5 \]
Thus, the height of the golf ball when it is 15 feet away from Lorraine is \( \boxed{22.5} \) feet.