approximately 13.63 ft.
Since we have a geometric series, the basic function for the arc length is:
f(x) = N*R^x
where
N = constant value for initial amplitude
R = Ratio of successive swings
x = swing number
Since we've been given the lengths of 2 different swings, let's express them as equations:
12 = N*R^7
20 = N*R^3
Now let's divide the 1st equation by the second, getting
12/20 = (N*R^7)/(N*R^3)
The N terms can be canceled, both top and bottom, giving
12/20 = R^7/R^3
And we can simply subtract the exponents to perform the division. So we have
12/20 = R^4
0.6 = R^4
0.880111737 = R
Now let's calculate N.
20 = N*R^3
20 = N*0.880111737^3
20 = N * 0.68173162
29.33705789 = N
So our function is:
f(x) = 29.33705789 * 0.880111737^x
Let's verify for the given values of 3 and 7.
f(3) = 29.33705789 * 0.880111737^3
f(3) = 29.33705789 * 0.68173162
f(3) = 20
f(7) = 29.33705789 * 0.880111737^7
f(7) = 29.33705789 * 0.409038972
f(7) = 12
Now let's calculate f(6)
f(6) = 29.33705789 * 0.880111737^6
f(6) = 29.33705789 * 0.464758002
f(6) = 13.6346324
So the length of the arc on the sixth swing is approximately 13.63 ft.
