Question

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

Answer 1

Answer:

  13.6 ft

Step-by-step 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