Question

A child is given an initial push on a rope swing. On the first? swing, the rope swings through an arc of

6

feet. On each successive? swing, the length of the arc is

90?%

of the previous length. After

15

?swings, what total length will the rope have? swung? When the child stops? swinging, what total length will the rope have? swung?

Answer #1

Given, on the first swing, the rope swings through an arc of 6 feet and on each successive swing, the length of the arc is 90% of the previous length.

Then, after 15 swings, the total length the rope will have swung
is = [6+6*(0.9)+6*(0.9)^{2}+....+6*(0.9)^{14}]
feet

= [6*(1-0.9^{15})/(1-0.9)] feet

47.65 feet

Now, after n swings, the total length the rope will have swung
is = [6*(1-0.9^{n})/(1-0.9)] feet

= [60*(1-0.9^{n})] feet

when n
, then 60*(1-0.9^{n})
60

Therefore, when the child stop swinging, then the total length the rope will have swung is 60 feet.

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