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 arc is 22 ft. long and the seventh arc is 10 ft. long, what is the length of the arc on the sixth swing? Round your answer to the nearest tenth of a foot. (1 point) 12.2ft. 13.0 ft. 14.8升t. 18.1 ft.

Answer 1

Let the common ratio be r. Then a3 = a1 r^2 = 22 and a7 = a1 r^6 = 10, so r^4 = a7/a3 = 10/22 = 5/11. The sixth arc is a6 = a3 r^3 = 22·r^3 = 22·(r^4)^{3/4} = 22·(5/11)^{3/4} ≈ 12.2 ft.

Answer: 12.2 ft.