Question

During halftime of a basketball game, a sling shot launches t-shirts into the crowd. The function h=−16t2+32t+5 models the t-shirt's height h in feet after t seconds. Use the vertex formula to determine how many seconds it will take a t-shirt to reach the maximum height. Then find the maximum height of the t-shirt. Reference Sheet - Features of a Parabola (2 points) A t-shirt will reach its maximum height at seconds. The maximum height of a t-shirt will be feet.

Answer 1

To find the time it takes for the t-shirt to reach its maximum height, we can use the vertex formula for a quadratic function of the form \( h(t) = at^2 + bt + c \). The time \( t \) at which the maximum height occurs can be found using the formula:

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

In the given function \( h(t) = -16t^2 + 32t + 5 \):
- \( a = -16 \)
- \( b = 32 \)

Now, we can plug these values into the vertex formula:

\[
t = -\frac{32}{2 \times -16} = -\frac{32}{-32} = 1
\]

So, the t-shirt will reach its maximum height at \( t = 1 \) second.

Next, we need to find the maximum height by substituting \( t = 1 \) back into the height function:

\[
h(1) = -16(1)^2 + 32(1) + 5
\]

Calculating this:

\[
h(1) = -16(1) + 32(1) + 5 = -16 + 32 + 5 = 16 + 5 = 21
\]

Thus, the maximum height of the t-shirt is 21 feet.

Summarizing the results:

- A t-shirt will reach its maximum height at **1 second**.
- The maximum height of a t-shirt will be **21 feet**.