During halftime of a football ​game, a sling shot launches​ T-shirts at the crowd. A​ T-shirt is launched from a height of 5 feet with an initial upward velocity of 64 feet per second. The​ T-shirt is caught 37 feet above the field. 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?

It takes 2 seconds to reach a maximum height of 69 feet, and the range is [5, 69].

The equation is of the form
h(t) = -16t² + v₀t + h₀, where -16 is the gravitational constant, v₀ is the initial velocity, and h₀ is the initial height.  Using the values from our problem, we have:

h(t) = -16t² + 64t + 5

To find the maximum height, we find the vertex.  The first step in this is to find the axis of symmetry, which is given by -b/2a:

-64/2(-16) = -64/-32 = 2

This is our value for t, so it takes 2 seconds to reach the maximum.  Substituting this into our function, we have
h(2) = -16(2²) + 64(2) + 5 = -64 + 128 + 5 = 64 + 5 = 69

This is the maximum height.

The range of heights goes from 5 to 69, inclusive, or [5, 69].