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?

After 14 swings, total length = 57.35 feet.

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

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.