Question

Lenny hit a baseball in the air, and the model y=−0.0009x2+0.3x+6.2
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)
The ball will reach a maximum height of
feet.

Answer 1

To find the maximum height of the ball, we need to determine the vertex of the parabola represented by the equation \(y = -0.0009x^2 + 0.3x + 6.2\). The vertex of a parabola in the form \(y = ax^2 + bx + c\) is found at the \(x\)-coordinate given by the formula:

\[
x = -\frac{b}{2a}
\]

In this case, \(a = -0.0009\) and \(b = 0.3\). Plugging in these values:

\[
x = -\frac{0.3}{2 \times -0.0009}
\]

Calculating the denominator:

\[
2 \times -0.0009 = -0.0018
\]

Now substituting:

\[
x = -\frac{0.3}{-0.0018} = \frac{0.3}{0.0018}
\]

Calculating this gives:

\[
x \approx 166.67 \text{ feet}
\]

Now we will substitute this value of \(x\) back into the original equation to calculate 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 back:

\[
y \approx -0.0009(27777.78) + 0.3(166.67) + 6.2
\]

Calculating each term:

1. \(-0.0009 \times 27777.78 \approx -24.999\)
2. \(0.3 \times 166.67 \approx 50\)
3. Adding these values together:

\[
y \approx -24.999 + 50 + 6.2
\]
\[
y \approx 31.201
\]

Rounding to the nearest tenth:

The maximum height that the ball reaches is approximately \(31.2\) feet.