Part A:
For this case we have a function of the form:
[tex]y = A * (b) ^ x
[/tex]
Where,
A: initial height
b: decrease rate
x: number of bounces
Substituting values we have:
[tex]y = 25 * (2/5) ^ x
[/tex]
Rewriting:
[tex]y = 25 * (0.40) ^ x [/tex]
Answer:
[tex]y = 25 * (0.40) ^ x
[/tex]
Part b:
See attached image
Part c:
By the time the height is less than 1 inch we have:
[tex]25 (0.40) ^ x \ \textless \ 1
[/tex]
From here, we clear x:
[tex](0.40) ^ x \ \textless \ 0.04
[/tex]
Logarithm on both sides:
[tex]x * log (0.40) \ \textless \ log (0.04)
[/tex]
Rewriting:
[tex]x * (- 1.39794) \ \textless \ (-0.39794)
[/tex]
[tex]x\ \textgreater \ (-0.39794) / (-1.39794)
[/tex]
[tex]x\ \textgreater \ 3.513
[/tex]
Answer:
x> 3.513 or x = 4
