A tire swing attached to a tree branch swings with an arc length of 30 feet on its first swing. On each swing following the first,

Question

A tire swing attached to a tree branch swings with an arc length of 30 feet on its first swing. On each swing following the first, the tire's arc length is 6/7of the length of the previous swing. Let an represent the arc length on the nth swing.

A) Find a formula that describes an. Express the common ratio as a fraction. Hint: Use an = a1r^n-1 and place r in parentheses!

an = ________________

B) What is the arc length of the tire on the 11th swing? Round your answer to three decimal places.

________________ feet

C) How many swings does it take before the arc length is less than one foot? Round your answer to a whole number. Hint: You must use the strategy for solving an exponential equation!

_________________

Answer 1

A) Simply plug in a_n = a₁ r^n-1 = 30 (6/7)^n-1.

B) a₁₁ = 30 (6/7)¹⁰ = 30 (0.21406) = 6.422 ft.

C) Set 1 = 30 (6/7)^n-1, rewrite as (1/30) = (6/7)^n-1 and take logarithms of both sides using the power rule (the base doesn't matter, so I use 10): log(1/30) = (n-1) log(6/7), n-1 = log(1/30) / log(6/7) = (-1.477) / (-0.0669) = 22.1. So n > 23.1 resulting in at least 24 swings.