Answer:
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.
