Question

During halftime of a soccer ​game, a sling shot launches​ T-shirts at the crowd. A​ T-shirt is launched from a height of 4 feet with an initial upward velocity of 80 feet per second. Use the equation  h left parenthesis t right parenthesis equals negative 16 t squared plus 80 t plus 4​, 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?
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Part 1
The​ T-shirt takes
  
enter your response here ​second(s) to reach its maximum height.
​(Type an integer or a​ decimal.)
The​ T-shirt's maximum height is
  
enter your response here feet above the field.
​(Type an integer or a​ decimal.)

Answer 1

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

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

In your equation:

\[
h(t) = -16t^2 + 80t + 4
\]

Here, \( a = -16 \) and \( b = 80 \).

Now, we can substitute these values into the formula:

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

So, the T-shirt takes **2.5 seconds** to reach its maximum height.

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

\[
h(2.5) = -16(2.5)^2 + 80(2.5) + 4
\]

Calculating \( (2.5)^2 \):

\[
(2.5)^2 = 6.25
\]

Then:

\[
h(2.5) = -16(6.25) + 80(2.5) + 4
\]
\[
= -100 + 200 + 4
\]
\[
= 100 + 4
\]
\[
= 104
\]

Thus, the T-shirt's maximum height is **104 feet** above the field.

### Summary
- The T-shirt takes **2.5 seconds** to reach its maximum height.
- The T-shirt's maximum height is **104 feet** above the field.