The function H (t) = - 16t2 + 64t + 12 shows the height H (t), in feet, of a baseball after t seconds. A second baseball moves in the air along a path represented by g (t) = 10 + 10.4t, where g (t) is the height, in feet, of the object from the ground at time t seconds.

Create a table using integers 1 through 4 for the 2 functions. Between what 2 seconds is the solution to H (t) = g (t) located? How do you know?

Explain what the solution from Part A means in the context of the problem.

We are given the two functions:

H (t) = - 16 t^2 + 64 t + 12

g (t) = 10 + 10.4 t

Now let us calculate for values of H (t) and g (t) by plugging in the values of t = 1 to 4 in each function as shown in the table below:

t H (t) g (t)

1 60 20.4

2 76 30.8

3 60 41.2

4 12 51.6

Looking at the table, we can say that the time when H (t) = g (t) must be between t=3 and t = 4 seconds.

At t=3 seconds H (t) is above g (t), and at t = 4 seconds H (t) is below g (t). Therefore the two functions must be at the same height in between.

Part B. The solution in Part A gives us the exact time where the two objects have the same height. Further, we can see from the table that H (t) is decreasing meaning the projectile is going down while g (t) is increasing so the other projectile is going up.