Question
Lenny hit a baseball in the air, and the model y=−0.0009x2+0.3x+6.2%0D%0A𝑦%0D%0A=%0D%0A−%0D%0A0.0009%0D%0A𝑥%0D%0A2%0D%0A+%0D%0A0.3%0D%0A𝑥%0D%0A+%0D%0A6.2%0D%0A represents the path of the ball, where x is the distance of the ball from where he hit (in feet) and y is the height reached by the ball (in feet). What is the maximum height that the ball reaches? Round to the nearest tenth.(1 point)
Answers
Answered by
GPT-4o mini
To find the maximum height of the ball represented by the quadratic function \( y = -0.0009x^2 + 0.3x + 6.2 \), we use the formula for the vertex of a parabola given in the format \( y = ax^2 + bx + c \), where \( a < 0 \) indicates that the parabola opens downwards and has a maximum point.
The x-coordinate of the vertex (which gives the distance at which the maximum height occurs) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
In this equation:
- \( a = -0.0009 \)
- \( b = 0.3 \)
Substituting these values in:
\[
x = -\frac{0.3}{2 \times -0.0009} = -\frac{0.3}{-0.0018} = \frac{0.3}{0.0018} \approx 166.67 \text{ feet}
\]
Now, we substitute \( x \) back into the original equation to find the maximum height \( y \):
\[
y = -0.0009(166.67)^2 + 0.3(166.67) + 6.2
\]
Calculating \( (166.67)^2 \):
\[
(166.67)^2 \approx 27777.78
\]
Now substitute this into the equation:
\[
y \approx -0.0009(27777.78) + 0.3(166.67) + 6.2
\]
Calculating each term:
\[
-0.0009(27777.78) \approx -24.999999 \approx -25
\]
\[
0.3(166.67) \approx 50.001
\]
Now sum everything up:
\[
y \approx -25 + 50.001 + 6.2 \approx 31.201
\]
Rounding to the nearest tenth gives:
\[
y \approx 31.2
\]
Thus, the maximum height that the ball reaches is **31.2 feet**.
The x-coordinate of the vertex (which gives the distance at which the maximum height occurs) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
In this equation:
- \( a = -0.0009 \)
- \( b = 0.3 \)
Substituting these values in:
\[
x = -\frac{0.3}{2 \times -0.0009} = -\frac{0.3}{-0.0018} = \frac{0.3}{0.0018} \approx 166.67 \text{ feet}
\]
Now, we substitute \( x \) back into the original equation to find the maximum height \( y \):
\[
y = -0.0009(166.67)^2 + 0.3(166.67) + 6.2
\]
Calculating \( (166.67)^2 \):
\[
(166.67)^2 \approx 27777.78
\]
Now substitute this into the equation:
\[
y \approx -0.0009(27777.78) + 0.3(166.67) + 6.2
\]
Calculating each term:
\[
-0.0009(27777.78) \approx -24.999999 \approx -25
\]
\[
0.3(166.67) \approx 50.001
\]
Now sum everything up:
\[
y \approx -25 + 50.001 + 6.2 \approx 31.201
\]
Rounding to the nearest tenth gives:
\[
y \approx 31.2
\]
Thus, the maximum height that the ball reaches is **31.2 feet**.