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?

Question

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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.