Given the scenario, we know that Lorraine hits the golf ball and it reaches a maximum height of 30 feet (the vertex of the parabola) when it is halfway between where it started and where it landed.
Step 1: Determine the vertex.
- The ball lands 60 feet away from Lorraine, so it started at 0 feet.
- The axis of symmetry of the parabola is halfway between the starting point and the landing point: \[ \text{Axis of Symmetry} = \frac{0 + 60}{2} = 30 \text{ feet} \]
- Therefore, the vertex is at \( (30, 30) \), where the x-coordinate is the distance from Lorraine and the y-coordinate is the height.
Step 2: Determine the equation of the parabola.
The general form of a parabola that opens downwards is: \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex. Here, \((h, k) = (30, 30)\).
So, we have: \[ f(x) = a(x - 30)^2 + 30 \]
Step 3: Find the value of \( a \).
We know the ball lands at a height of 0 feet when it is 60 feet away from Lorraine. Therefore, we can substitute \( x = 60 \) and \( f(60) = 0 \) into the equation to find \( a \): \[ 0 = a(60 - 30)^2 + 30 \] \[ 0 = a(30)^2 + 30 \] \[ 0 = 900a + 30 \] \[ 900a = -30 \] \[ a = -\frac{1}{30} \]
Step 4: Write the complete equation.
Thus, the equation for the scenario can now be written as: \[ f(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
Step 5: Calculate the height of the ball when it is 15 feet away from Lorraine.
Now to find the height when \( x = 15 \): \[ f(15) = -\frac{1}{30}(15 - 30)^2 + 30 \] \[ = -\frac{1}{30}(-15)^2 + 30 \] \[ = -\frac{1}{30}(225) + 30 \] \[ = -\frac{225}{30} + 30 \] \[ = -7.5 + 30 \] \[ = 22.5 \]
Final answers:
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The equation is: \[ f(x) = -\frac{1}{30}(x - 30)^2 + 30 \]
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The height of the ball when it is 15 feet away from Lorraine is: \[ 22.5 \text{ feet in the air.} \]