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14 November, 02:04

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?

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  1. 14 November, 05:33
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    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.
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