Question

A child is given a push on a swing.Each time she swings back a little less, according to the equation d=2(0.85)^n , where d is the horizontal distance, in metre, from her position after n complete swings. After how many swings will the distance from rest be less than 15cm.

Answer 1

Ooo this is annoying.. so they're measuring d in meters.
And they want to know when d<15cm.

I hate these unit changes >:c

So they're really asking when d<.15meters, right?
Did I do that conversion correctly?

Answer 2

yes

Answer 3

So we let our distance from the horizontal be .15 meters,
\[\Large\rm .15=2(.85)^n\]And we need to solve for n, yes?

Looks like we'll need to use logs.

Answer 4

Divide each side by 2,\[\Large\rm .075=.85^n\]Then take the natural log of each side,\[\Large\rm \ln .075= \ln(.85^n)\]
The reason we're using logs issssss,
they help us get variables out of the exponent position.

Answer 5

yaa you are right

Answer 6

Rule of logs:\[\Large\rm \color{royalblue}{\log(a^b)=b \log (a)}\]Do you see how we can apply this rule to our problem here?

Answer 7

yes

Answer 8

Log (.85)^n = blog

Answer 9

(.075)

Answer 10

Ok great! The n comes out front as a multiplier.
\[\Large\rm \ln .075= n ~\ln.85\]
Understand how to solve for n from that point?

Answer 11

yes i understand thanks a lot

Answer 12

cool