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Sagot :
To analyze the trajectory of the golf ball, we will use the given parabolic equation for the height, [tex]\( h(x) = -0.25x^2 + 4.3x \)[/tex], where [tex]\( x \)[/tex] represents the horizontal distance in feet from the starting point.
1. The ball starts [tex]\(\square\)[/tex] feet above the ground:
This is the initial height of the ball, which occurs when [tex]\( x = 0 \)[/tex]. We can find it by substituting [tex]\( x = 0 \)[/tex] in the height equation:
[tex]\[ h(0) = -0.25(0)^2 + 4.3(0) = 0 \][/tex]
Therefore, the ball starts 0 feet above the ground.
2. The ball reaches a maximum height of [tex]\(\square\)[/tex] feet at a horizontal distance of [tex]\(\square\)[/tex] feet away from the golf club it was hit with:
The maximum height of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at its vertex. The x-coordinate of the vertex can be found by the formula:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the equation [tex]\( h(x) = -0.25x^2 + 4.3x \)[/tex]:
[tex]\[ a = -0.25, \quad b = 4.3 \][/tex]
Substituting these values in:
[tex]\[ x = \frac{-4.3}{2(-0.25)} = \frac{-4.3}{-0.5} = 8.6 \][/tex]
So, the maximum height occurs at [tex]\( x = 8.6 \)[/tex] feet. To find the corresponding height, substitute [tex]\( x = 8.6 \)[/tex] back into the height equation:
[tex]\[ h(8.6) = -0.25(8.6)^2 + 4.3(8.6) \][/tex]
[tex]\[ h(8.6) = -0.25(73.96) + 36.98 = -18.49 + 36.98 = 18.49 \][/tex]
Therefore, the maximum height is 18.49 feet, occurring at a horizontal distance of 8.6 feet.
3. The ball returns to the ground at about [tex]\(\square\)[/tex] feet away:
The ball returns to the ground when its height [tex]\( h(x) \)[/tex] becomes 0 again. We solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ -0.25x^2 + 4.3x = 0 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case:
[tex]\[ a = -0.25, \quad b = 4.3, \quad c = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] to find the roots:
[tex]\[ x = \frac{-4.3 \pm \sqrt{4.3^2 - 4(-0.25)(0)}}{2(-0.25)} \][/tex]
[tex]\[ x = \frac{-4.3 \pm \sqrt{18.49}}{-0.5} \][/tex]
Thus:
[tex]\[ x = \frac{4.3 \pm 4.3}{-0.5} \][/tex]
The roots are:
[tex]\[ x = \frac{4.3 + 4.3}{-0.5} = 17.2 \quad \text{and} \quad x = \frac{4.3 - 4.3}{-0.5} = 0 \][/tex]
Since the ball starts at [tex]\( x = 0 \)[/tex], the other root is the distance at which the ball returns to the ground, which is approximately 17.2 feet.
Summarizing the findings:
- The ball starts [tex]\(\boxed{0}\)[/tex] feet above the ground.
- The ball reaches a maximum height of [tex]\(\boxed{18.49}\)[/tex] feet at a horizontal distance of [tex]\(\boxed{8.6}\)[/tex] feet away from the golf club it was hit with.
- The ball returns to the ground at about [tex]\(\boxed{17.2}\)[/tex] feet away.
1. The ball starts [tex]\(\square\)[/tex] feet above the ground:
This is the initial height of the ball, which occurs when [tex]\( x = 0 \)[/tex]. We can find it by substituting [tex]\( x = 0 \)[/tex] in the height equation:
[tex]\[ h(0) = -0.25(0)^2 + 4.3(0) = 0 \][/tex]
Therefore, the ball starts 0 feet above the ground.
2. The ball reaches a maximum height of [tex]\(\square\)[/tex] feet at a horizontal distance of [tex]\(\square\)[/tex] feet away from the golf club it was hit with:
The maximum height of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at its vertex. The x-coordinate of the vertex can be found by the formula:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the equation [tex]\( h(x) = -0.25x^2 + 4.3x \)[/tex]:
[tex]\[ a = -0.25, \quad b = 4.3 \][/tex]
Substituting these values in:
[tex]\[ x = \frac{-4.3}{2(-0.25)} = \frac{-4.3}{-0.5} = 8.6 \][/tex]
So, the maximum height occurs at [tex]\( x = 8.6 \)[/tex] feet. To find the corresponding height, substitute [tex]\( x = 8.6 \)[/tex] back into the height equation:
[tex]\[ h(8.6) = -0.25(8.6)^2 + 4.3(8.6) \][/tex]
[tex]\[ h(8.6) = -0.25(73.96) + 36.98 = -18.49 + 36.98 = 18.49 \][/tex]
Therefore, the maximum height is 18.49 feet, occurring at a horizontal distance of 8.6 feet.
3. The ball returns to the ground at about [tex]\(\square\)[/tex] feet away:
The ball returns to the ground when its height [tex]\( h(x) \)[/tex] becomes 0 again. We solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ -0.25x^2 + 4.3x = 0 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case:
[tex]\[ a = -0.25, \quad b = 4.3, \quad c = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] to find the roots:
[tex]\[ x = \frac{-4.3 \pm \sqrt{4.3^2 - 4(-0.25)(0)}}{2(-0.25)} \][/tex]
[tex]\[ x = \frac{-4.3 \pm \sqrt{18.49}}{-0.5} \][/tex]
Thus:
[tex]\[ x = \frac{4.3 \pm 4.3}{-0.5} \][/tex]
The roots are:
[tex]\[ x = \frac{4.3 + 4.3}{-0.5} = 17.2 \quad \text{and} \quad x = \frac{4.3 - 4.3}{-0.5} = 0 \][/tex]
Since the ball starts at [tex]\( x = 0 \)[/tex], the other root is the distance at which the ball returns to the ground, which is approximately 17.2 feet.
Summarizing the findings:
- The ball starts [tex]\(\boxed{0}\)[/tex] feet above the ground.
- The ball reaches a maximum height of [tex]\(\boxed{18.49}\)[/tex] feet at a horizontal distance of [tex]\(\boxed{8.6}\)[/tex] feet away from the golf club it was hit with.
- The ball returns to the ground at about [tex]\(\boxed{17.2}\)[/tex] feet away.
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