During halftime of a football ​game, a sling shot launches​ T-shirts at the crowd. A​ T-shirt is launched from a height of 5 feet with an initial upward velocity of 88 feet per second. Use the equation h left parenthesis t right parenthesis equals negative 16 t squared plus 88 t plus 5​, 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, which is given by:

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

In the equation \(h(t) = -16t^2 + 88t + 5\):

• \(a = -16\)
• \(b = 88\)

Substituting these values into the vertex formula:

\[ t = -\frac{88}{2(-16)} = -\frac{88}{-32} = \frac{88}{32} = 2.75 \]

So, the T-shirt takes 2.75 seconds to reach its maximum height.

Next, we need to find the maximum height by substituting \(t = 2.75\) back into the height equation \(h(t)\):

\[ h(2.75) = -16(2.75)^2 + 88(2.75) + 5 \]

Calculating each term:

1. \( (2.75)^2 = 7.5625 \)
2. \( -16(7.5625) = -121 \)
3. \( 88(2.75) = 242 \)
4. Combining these:

\[ h(2.75) = -121 + 242 + 5 = 126 \]

Therefore, the T-shirt's maximum height is 126 feet above the field.

To summarize:

• The T-shirt takes 2.75 seconds to reach its maximum height.
• The T-shirt's maximum height is 126 feet above the field.