Question

During halftime of a basketball game a slingshot launches t shirts at the crowd. A T-shirt is launched from a height of 4 feet with an initial upward velocity of 88 feet per second. The T-shirt is caught 45 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?

Answer 1

Normally in these type of questions, air resistance can be ignored, but launching a t-shirt ????

anyway ...
height = -16t^2 + 88t + 4

you want the vertex of the corresponding parabola
the x of the vertex is -88/(2(-16)) = 11/4
the corresponding height is -16(11/4)^2 + 88(11/4) + 4 = 125

but it was caught at a height of 45 ft (assuming on its way up)
so 45 = -16t^2 + 88t + 4
16t^2 - 88t + 41 = 0

solve using your favourite method of solving quadratics
and then interpret your result

Answer 2

Old book I guess. Assume a here = -32 ft/s^2 which is g in old units
h = 4 + 88 t - (32/2) t^2
v = 88 - 32 t
at max height v = 0
so t = 88/32 = 2.75 seconds
then at t = 2.75
h = 4 + 88 (2.75) - 16 (2.75)^2
= 4 + 242 - 121 = 125 feet at very top
now I suppose the T shirt is caught on the way down
at the top , h = 125, the speed is zero
so this is really something dropped from 125 ft and caught at 45 feet
fall 125 - 45 = 80 feet so you could calculate when it is caught.
Now the range is from a height of 4 feet to a height of 125 feet