During halftime of a basketball 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. The T-shirt is caught 38 feet above the court. 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

During halftime of a basketball 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. The T-shirt is caught 38 feet above the court. 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?

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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.)

Answer 1

The T-shirt takes 2 seconds to reach its maximum height.

Part 2
The maximum height can be found using the equation for vertical motion:
h(t) = -16t^2 + v0t + h0

where h(t) is the height of the T-shirt at time t, v0 is the initial upward velocity (72 ft/s), and h0 is the initial height (3 ft).

Plugging in the values:
h(2) = -16(2)^2 + 72(2) + 3
h(2) = -64 + 144 + 3
h(2) = 83 ft

So, the maximum height the T-shirt reaches is 83 feet.

Part 3
The range of the function that models the height of the T-shirt over time is from t=0 to the time it takes for the T-shirt to reach its maximum height, which is t=2 seconds. Therefore, the range is 0 ≤ t ≤ 2.