Exponential functions

Question

A ball dropped from 20m bounces to a height of 5/8 of its previous height. Write an exponential function f(b), modeling the balls height after b bounces and use it to determine the height of the fifth bounce to the nearest centimeter

show each step

Answer 1

Alyssa,

Im eager to show you how, rather than just do it. So first, you’ve got to figure out what is the assignment is.

It looks like the assignment is:

1) write a function

2) use the function

So in more detail, “write the function”: I can give a hint about how to do that. I’d start by writing a really easy function that tells the height of one bounce. I’ll call it f(1). Then I’d figure how to change it into a function that handles b, any number of bounces.

Translate “A ball dropped from 20m bounces to a height of 5/8 of its previous height” into

f(1) = 20*(5/8)

so here’s a question: how would you write down the height after two bounces?

I can see an easy way and a tedious way. The tedious way is to calculate out 20*5/8, and then multiply by 5/8 for the second time.

f(1) = 20*5/8 = 100/8 = 25/2;

f(2) = 25/2 * 5/8 = (25*5)/(2*8) = 125/16

The easy way is to write down both 5/8 at once, and leave them alone.

f(2) = 20*(5/8)*(5/8)

So what would f(3) look like? Just multiply f(2) by 5/8

f(3) = 20*(5/8)*(5/8)*(5/8)

Do you see a pattern? Replace all those 5/8 s with one 5/8 and an exponent:

f(3) = 20*(5/8)³

So If b is the number of bounces, then

f(b) = 20*(5/8)^b

Your function is complete. I hope I gave a hint about one of many ways to think about a recursive functions.

Now all you have to do is evaluate your function for 5 bounces.

f(5) = 20*(5/8)⁵

This is trivial with a calculator

(5/8)⁵ = .09536

f(5) = 20*(5/8)⁵ = 20*.09536 = 1.9072 m = 1.91 to nearest centimeter