Question

The flight of a baseball is modeled by the function h=−16t2+80t+3
ℎ
=
−
16
𝑡
2
+
80
𝑡
+
3
where h is the height of the baseball above the ground after t seconds.

Use the vertex formula to determine how many seconds it will take the baseball to reach the maximum height.

Then find the maximum height of the baseball.

Reference Sheet - Features of Parabola

(2 points)
The baseball will reach its maximum height at seconds.

The maximum height of the baseball will be feet.

Answer 1

To find the time at which the baseball reaches its maximum height using the vertex formula, we can use the formula for the vertex of a parabola given by:

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

In the equation \( h = -16t^2 + 80t + 3 \), we identify the coefficients as follows:

- \( a = -16 \)
- \( b = 80 \)

Now, substituting these values into the vertex formula:

\[
t = -\frac{80}{2(-16)} = -\frac{80}{-32} = \frac{80}{32} = 2.5
\]

Therefore, the baseball will reach its maximum height at **2.5 seconds**.

Next, we substitute \( t = 2.5 \) back into the height equation to find the maximum height:

\[
h = -16(2.5)^2 + 80(2.5) + 3
\]

Calculating \( (2.5)^2 \):

\[
(2.5)^2 = 6.25
\]

Now substituting this value into the height equation:

\[
h = -16(6.25) + 80(2.5) + 3
\]
\[
h = -100 + 200 + 3
\]
\[
h = 100 + 3 = 103
\]

Therefore, the maximum height of the baseball will be **103 feet**.

In summary:

- The baseball will reach its maximum height at **2.5 seconds**.
- The maximum height of the baseball will be **103 feet**.