Answer:
The height of the bouncing ball appears to be decreasing by a factor of one-half each time it bounces. We can model this situation with an exponential decay function of the form:
h(t) = a * (1/2)^t
where h(t) represents the height of the ball at time t, a represents the initial height of the ball (in inches), and t represents the number of bounces.
Using the given data, we can set up a system of equations to solve for a:
h(0) = a = 54 (initial height)
h(1) = a * (1/2)^1 = 36 (height at second peak)
h(2) = a * (1/2)^2 = 24 (height at third peak)
Solving for a, we get:
a = h(0) = 54
a * (1/2)^1 = 36
a * (1/2)^2 = 24
Multiplying the second equation by 2 and the third equation by 4, we can eliminate the fractions:
a = 54
a * 1 = 72
a * 4 = 96
Solving for a in the first equation, we get a = 54. Therefore, the function that represents the height of the ball after t bounces is:
h(t) = 54 * (1/2)^t
To generate a sequence from the formula f(x + 1) = One-half(f(x)), we can start with an initial value for f(0) and use the formula to generate subsequent terms. In this case, we can use the formula with f(0) = 1 to generate the sequence:
f(1) = 1/2
f(2) = 1/4
f(3) = 1/8
f(4) = 1/16
and so on.
Each term in the sequence is half the value of the previous term, so the sequence is decreasing exponentially.