Answer:
- time to max height: 2.5 seconds
- max height: 103 ft
- range: [3, 103] ft
Step-by-step explanation:
Ballistic motion is usually modeled by the quadratic function ...
h(t) = -1/2g·t^2 +v0·t +h0
where g is the acceleration due to gravity, v0 is the initial upward velocity, and h0 is the initial height.
When units are in feet and seconds, the value of g is usually taken to be 32 ft/s^2, so for v0 = 80 ft/s and h0 = 3 ft, the equation is ...
h(t) = -16t^2 +80t +3
The axis of symmetry of the graph of this equation is ...
t = v0/g = (80 ft/s)/(32 ft/s^2)
t = 2.5 s . . . . . the time to reach the maximum height
At that time, the maximum height is ...
h(2.5) = (-16·2.5 +80)2.5 +3 = 103 . . . . feet
Then the range of the function extends from its minimum value, 3 at t=0, to its maximum value, 103 at t=2.5. Since the T-shirt is caught at h(t) = 43, the T-shirt never goes below 3 feet.
The range of the function is 3 ft to 103 ft.
_____
Alternate solution
The graph suggests an alternate solution. Since the T-shirt is caught at a height of 43 ft, this could be the maximum height the T-shirt reaches. That will occur at about t=0.564 seconds. Then the range of the function is 3–43 feet.
