About a new softball's bounce height...

Question

A new softball dropped onto a hard surface from a height of 25 inches rebounds to about 2/5 the height on each successive bounce. 

1. Write a function representing the rebound height for each bounce.

2. After how many bounces would a new softball rebound less than 1 inch?

Answer 1

William,

 

Let    h_o = original height ball dropped from (= 25 in.)

        h_n = height reached after bounce n = 0, 1, 2, 3 ...

 

Then h_n = h_o (0.40)ⁿ  ,  where 0.40 = 2/5  

 

How many bounces will it take for h_n < 1 in?

h_o(0.40)ⁿ = 25 (0.40)ⁿ < 1  or (0.40)ⁿ < 0.04  

Taking log to base 10 of both sides

n log (0.40) < log (0.04)

n (-1.39794) < (-0.39794)

n > (-0.39794) / (-1.39794)  since dividing by negative number flips the < relation to >

n > 3.513 

or n = 4

 

Check:  if n=3, h₃ = 25(0.40)³ = 25 (0.064)    = 1.6 in.

            if n=4, h₄ = 25(0.40)⁴ = 25 (0.0256) = 0.64 in. so checks . NOTE also that h₄ = (0.4) h₃.