Question

A rope is swinging In such a way that the length of the arc traced by a knot at its bottom end is

decreasing geometrically. If the third are is 20 ft. long and the seventh arc is 12 ft. long, what is
the sixth swing? Round your answer
the length of the arc on
to the nearest tenth of a foot.

Answer 1

13.6 ft

Explanation:

The geometric sequence of arc lengths can be described by ...

f(n) = a·b^n

We have (n, f(n)) = (3, 20) and (7, 12). Using these values, we can find the common ratio (b):

20 = a·b^3

12 = a·b^7

Then ...

12/20 = (a·b^7)/(a·b^3) = b^4 = 3/5

We want the 6th term, which we can get from the 7th term by multiplying by b^-1.

b^(-1) = (b^4)^(-1/4) = (3/5)^(-1/4) = √(√(5/3)) ≈ 1.13622

Then the 6th swing had an arc length of ...

f(6) = f(7)·b^-1

f(6) = (12 ft)(1.13622) ≈ 13.63 ft ≈ 13.6 ft