h = -15 sin (π/7 (t-2)) + 42

h is the height of the ball above the floors in inches at time t seconds

1) what is the highest the ball will ever bounce?
2) when is the first time the ball bounces that high?
3) write a general expression for the times at which the ball is at its highest point.

(1) surely you know that the maximum value of sin() is 1. So, 15+42 is the max for h

(2) you want sin (π/7 (t-2)) to be -1, so
(π/7 (t-2)) = 3π/2
t=25/2

(3) since the period is 2π/(π/7) = 14, the max will occur at 25/2 + 14k for integer values of k.

The above assertions can be seen on the graph at

http://www.wolframalpha.com/input/?i=-15+sin+%28%CF%80%2F7+%28t-2%29%29+%2B+42+for+0%3C%3Dt%3C%3D20

1) To find the highest the ball will ever bounce, we need to determine the maximum value of the height function h. In this case, since h is given by h = -15 sin (π/7 (t-2)) + 42, we can see that the maximum value will occur when the sine function reaches its maximum value of 1. This happens when the argument of the sine function, π/7 (t-2), is equal to π/2.

To find t, we can set π/7 (t-2) = π/2 and solve for t:
π/7 (t-2) = π/2
t-2 = (π/2)(7/π)
t-2 = 7/2
t = 7/2 + 2
t = 11/2

Therefore, the highest the ball will ever bounce is at t = 11/2 seconds.

2) We have already determined in part 1 that the highest bounce occurs at t = 11/2 seconds. To find when the ball first bounces that high, we need to determine the first positive value of t for which h equals the maximum value.

Substituting t = 11/2 into the height function:
h = -15 sin (π/7 (11/2-2)) + 42
h = -15 sin (π/7 (11/2-4/2)) + 42
h = -15 sin (π/7 (11-4)/2) + 42
h = -15 sin (π/7 (7)/2) + 42
h = -15 sin (π/2) + 42
h = -15(1) + 42
h = 27

Therefore, the ball bounces to its maximum height of 27 inches for the first time at t = 11/2 seconds.

3) To write a general expression for the times at which the ball is at its highest point, we need to consider the periodic nature of the sine function. The sine function repeats itself every 2π units, so the general expression for t can be written as:

t = 11/2 + (2π/π/7)n

where n is an integer representing the number of full periods completed after the first highest bounce at t = 11/2 seconds.