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 64 feet per second. The T-shirt is caught 43 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 64 feet per second. The T-shirt is caught 43 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 content area bottom
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 approximately 1 second to reach its maximum height.

Part 2
The maximum height of the T-shirt can be calculated using the formula for projectile motion:
h(t) = h0 + v0t - (1/2)gt^2
where:
h(t) = height at time t
h0 = initial height (3 feet)
v0 = initial velocity (64 feet per second)
g = acceleration due to gravity (32 feet per second squared)

Plugging in the values:
h(t) = 3 + 64t - 16t^2

To find the maximum height, we need to find the vertex of the quadratic function. The formula for the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is:
x = -b / 2a

In this case, a = -16, b = 64, c = 3
x = -64 / (2*(-16))
x = 2 seconds

Plugging in t = 2 seconds into the height function:
h(2) = 3 + 64(2) - 16(2)^2
h(2) = 3 + 128 - 64
h(2) = 67 feet

Therefore, the maximum height of the T-shirt is 67 feet.

Part 3
The range of the function that models the height of the T-shirt over time can be found by looking at the x-intercepts of the function. Since the T-shirt reaches its maximum height at 2 seconds and falls back to the ground, the range of the function is from 0 seconds to 2 seconds.

Answer 2

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 64 feet per second. The T-shirt is caught 43 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 content area bottom
Part 1
The T-shirt takes

2 second(s) to reach its maximum height.
(Type an integer or a decimal.)
Part 2
The T-shirt's maximum height is

enter your response here feet above the court.
(Type an integer or a decimal.)

Answer 3

The T-shirt's maximum height is 67 feet above the court.

Answer 4

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 64 feet per second. The T-shirt is caught 43 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 content area bottom
Part 1
The T-shirt takes

2 second(s) to reach its maximum height.
(Type an integer or a decimal.)
Part 2
The T-shirt's maximum height is

67 feet above the court.
(Type an integer or a decimal.)
Part 3
What is the range of the function?
A.
3less than or equalshless than or equals67
B.
All real numbers
C.
43less than or equalshless than or equals67
D.
0less than or equalshless than or equals67

Answer 5

The range of the function that models the height of the T-shirt over time is:

C. 43 ≤ h ≤ 67

This means that the height of the T-shirt will always be between 43 feet and 67 feet above the court.