Question

During halftime of a football ​game, a sling shot launches​ T-shirts at the crowd. A​ T-shirt is launched from a height of 3 feet with an initial upward velocity of 72 feet per second. Use the equation h left parenthesis t right parenthesis equals negative 16 t squared plus 72 t plus 3​, where t is time in seconds and ​h(t) is height. How long will it take the​ T-shirt to reach its maximum​ height? What is the maximum​ height?Question content area bottomPart 1The​ T-shirt takesenter your response here ​second(s) to reach its maximum height.​(Type an integer or a​ decimal.)The​ T-shirt's maximum height isenter your response here feet above the field.

Answer 1

To determine when the T-shirt reaches its maximum height, we can analyze the given quadratic function:

\[
h(t) = -16t^2 + 72t + 3
\]

### Part 1: Time to Reach Maximum Height

The maximum height of a parabola described by the equation \( h(t) = at^2 + bt + c \) occurs at the vertex. The formula for the vertex's x-coordinate (in this case, time \( t \)) is given by:

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

Here, \( a = -16 \) and \( b = 72 \):

\[
t = -\frac{72}{2(-16)} = \frac{72}{32} = \frac{9}{4} = 2.25 \text{ seconds}
\]

### Part 2: Maximum Height

Next, we will use this time to find the maximum height by substituting \( t = 2.25 \) back into the height function \( h(t) \):

\[
h(2.25) = -16(2.25)^2 + 72(2.25) + 3
\]

Calculating \( (2.25)^2 \):

\[
(2.25)^2 = 5.0625
\]

Now substituting back into the height equation:

\[
h(2.25) = -16(5.0625) + 72(2.25) + 3
\]

Calculating each term:

\[
-16(5.0625) = -81 \\
72(2.25) = 162
\]

Now putting everything together:

\[
h(2.25) = -81 + 162 + 3 = 84 \text{ feet}
\]

### Final Answers

- The T-shirt takes **2.25 seconds** to reach its maximum height.
- The T-shirt's maximum height is **84 feet** above the field.