Question

A child is given an initial push on a rope swing. On the first​ swing, the rope swings through an arc of 12 feet. On each successive​ swing, the length of the arc is 80​% of the previous length. After 14 ​swings, what total length will the rope have​ swung? When the child stops​ swinging, what total length will the rope have​ swung?

Answer 1

After 14 swings, total length = 57.35 feet.

When the child stops swinging, Total length = 60 feet.

Step-by-step explanation:

The total length of the rope after 14 swings form a geometric progression which is also known as exponential sequence.

The sum of the term in a Geometry progression is

Sₙ = a(1-rⁿ)/1-r.................(1)

Where Sₙ = sum of the nth term, a= first term, n= number of term, r= common ratio.

n=14, a= 12 feet, r=80% = 0.8.

Substituting the values above into equation(1)

S₁₄ = 12(1-0.8¹⁴)/1-0.8

S₁₄= 12(1-0.04398)/1-0.8

S₁₄= 12(0.95602)/0.2

S₁₄ = 11.47/0.2

S₁₄= 57.35 feet

∴ After 14 swings, the total length the rope will swing is = 57.35 feet.

The total length of the rope when the child stop swinging = sum to infinity of the Geometry progression( exponential sequence).

The sum to infinity of an exponential sequence

S = a/1-r

Where a= first term= 12 feet, r= common ratio = 0.8.

∴ S= 12/1-0.8

S= 12/0.2

S = 60 feet

When the boy stops swinging, the total length the rope have swung = 60 feet.