During the halftime of a basketball game, a sling shot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocity of 72 ft/s. The t-shirt is caught 35 ft above the court. The function y= -16t^2 + 72t +5 gives the T-shirts height h, in feet, after t seconds.
a. How long will it take the T-shirt to reach its maximum height?
b. What is the maximum height?
c. What is the range of the function that models the height of the T-shirt over time?
A) The T-shirt will be at its maximum height when the velocity is at zero, because its velocity will be at zero when it reaches the top of its arc.
You have the position function, y= -16t^2 + 72t + 5, so the velocity function is its derivative, v= -32t + 72.
Set that equal to zero, 0= -32t + 72, and solve for t.
0= -32t + 72
32t = 72
t = 9/4 or 2.25 seconds.
b) Plug what you found in part a) back into your position equation.
-16(2.25)^2 + 72(2.25) + 5 = -81 + 162 + 5 = 86 feet
c) The t-shirt’s lowest point is 5 feet above the ground (that’s the ’+ 5’ in the position equation) and its highest point is 86 feet, which you found in part b). So the range is [5,86]