Let's analyze the given parabolic equation for the height of the golf ball: [tex]\( h(x) = -0.25x^2 + 3.4x \)[/tex], where [tex]\( x \)[/tex] represents the horizontal distance from the golf club in feet, and [tex]\( h(x) \)[/tex] represents the height of the ball in feet.
1. Initial Height:
The initial height of the ball can be determined by evaluating the height equation when [tex]\( x = 0 \)[/tex] (right at the moment when the ball starts from the ground).
[tex]\[
h(0) = -0.25(0)^2 + 3.4(0) = 0
\][/tex]
Thus, the ball starts 0 feet above the ground.
2. Maximum Height and Horizontal Distance:
The maximum height of the ball can be found by determining the vertex of the parabola. For the given quadratic equation [tex]\( h(x) = -0.25x^2 + 3.4x \)[/tex], the vertex form is used by finding the value of [tex]\( x \)[/tex] at the vertex:
[tex]\[
x_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
Here, [tex]\( a = -0.25 \)[/tex] and [tex]\( b = 3.4 \)[/tex]. Thus,
[tex]\[
x_{\text{vertex}} = -\frac{3.4}{2(-0.25)} = 6.8
\][/tex]
Plugging [tex]\( x_{\text{vertex}} \)[/tex] back into the height equation to find the maximum height:
[tex]\[
h(6.8) = -0.25(6.8)^2 + 3.4(6.8) = 11.56 \text{ feet}
\][/tex]
Thus, the ball reaches a maximum height of 11.6 feet at a horizontal distance of 6.8 feet away from the golf club.
3. Distance Where the Ball Returns to the Ground:
The ball returns to the ground when the height [tex]\( h(x) = 0 \)[/tex]. Solving the quadratic equation for [tex]\( x \)[/tex] when [tex]\( h(x) = 0 \)[/tex]:
[tex]\[
0 = -0.25x^2 + 3.4x
\][/tex]
Factoring out [tex]\( x \)[/tex]:
[tex]\[
0 = x(-0.25x + 3.4)
\][/tex]
This gives us two solutions:
[tex]\[
x = 0 \quad \text{or} \quad -0.25x + 3.4 = 0
\][/tex]
Solving the second equation for [tex]\( x \)[/tex]:
[tex]\[
-0.25x + 3.4 = 0 \implies x = \frac{3.4}{0.25} = 13.6
\][/tex]
Thus, the ball returns to the ground at about 13.6 feet away from the golf club.
In summary:
- The ball starts 0 feet above the ground.
- The ball reaches a maximum height of 11.6 feet at a horizontal distance of 6.8 feet away from the golf club it was hit with.
- The ball returns to the ground at about 13.6 feet away.
