During halftime of a soccer game, a sling shot launches T-shirts at the crowd. A T-shirt is launched from a height of 6 feet with an initial upward velocity of 88 feet per second. The T-shirt is caught 36 feet above the field. How long will it take the T-shirt to reach its maximum height? What is the maximum height? What is the range of the function that models the height of the T-shirt over time?Question content area bottomPart 1The T-shirt takesenter your response here second(s) to reach its maximum height.(Type an integer or a decimal.)

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 6 feet with an initial upward velocity of 88 feet per second. The​ T-shirt is caught 36 feet above the field. How long will it take the​ T-shirt to reach its maximum​ height? What is the maximum​ height? What is the range of the function that models the height of the​ T-shirt over​ time?Question content area bottomPart 1The​ T-shirt takesenter your response here ​second(s) to reach its maximum height.​(Type an integer or a​ decimal.)

GPT-4o mini · Answer

To determine how long it takes for the T-shirt to reach its maximum height and what that maximum height is, we can use the equations of motion for an object under the influence of gravity. The height $h (t)$ of the T-shirt at time $t$ seconds can be modeled by the following quadratic equation:

$h (t) = - 16 t^{2} + v_{0} t + h_{0}$

where:

$v_{0}$ is the initial velocity (88 ft/s),
$h_{0}$ is the initial height (6 ft),
$- 16 t^{2}$ is the effect of gravity (measured in feet per second squared, using the conversion factor for the height equation).

Plugging in the values, we have:

$h (t) = - 16 t^{2} + 88 t + 6$

Part 1: Time to reach maximum height

The time to reach the maximum height for a quadratic function $a x^{2} + b x + c$ can be found using:

$t = - \frac{b}{2 a}$

In our equation:

$a = - 16$
$b = 88$

So:

$t = - \frac{88}{2 \cdot - 16} = \frac{88}{32} = 2.75 seconds$

Part 2: Maximum height

Now, plug $t = 2.75$ back into the height formula to find the maximum height:

$h (2.75) = - 16 (2.75)^{2} + 88 (2.75) + 6$ $= - 16 (7.5625) + 242 + 6$ $= - 121 + 242 + 6$ $= 127 feet$

Part 3: Range of the function

To determine the range of the function, we need to identify the minimum and maximum heights that the T-shirt achieves.

The maximum height is found at $t = 2.75$ seconds and is $127$ feet.
The T-shirt is caught at a height of $36$ feet.

Now we need to find when it reaches $36$ feet. Set $h (t) = 36$ :

$- 16 t^{2} + 88 t + 6 = 36$

Rearranging gives us:

$- 16 t^{2} + 88 t + 6 - 36 = 0 ⟹ - 16 t^{2} + 88 t - 30 = 0$

Using the quadratic formula $t = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ :

$a = - 16$
$b = 88$
$c = - 30$

Calculating the discriminant:

$b^{2} - 4 a c = 88^{2} - 4 (- 16) (- 30)$ $= 7744 - 1920 = 5824$

Now, calculate the roots:

$t = \frac{- 88 \pm \sqrt{5824}}{2 \cdot - 16}$ Approximate the square root:

$\sqrt{5824} \approx 76.3$

Thus:

$t = \frac{- 88 \pm 76.3}{- 32}$

Calculating the two possible times:

$t_{1} = \frac{- 88 + 76.3}{- 32} \approx \frac{- 11.7}{- 32} \approx 0.3656 seconds$

$t_{2} = \frac{- 88 - 76.3}{- 32} \approx \frac{- 164.3}{- 32} \approx 5.14375 seconds$

Range of the function

Since the T-shirt is launched and caught, the range of the function is between the height of 6 feet (the initial height) and 127 feet (the maximum height). However, since it is caught at 36 feet, the range can be stated as from 6 feet up to 127 feet, noting that it is caught at 36 feet.

Thus, the final answers are:

It takes 2.75 seconds to reach maximum height.
The maximum height is 127 feet.
The range of the function is [6, 127] (considering the initial height and the maximum height).