Question

a rope is swinging in such a way that the length of the arc traced by a knot at its bottom is decreasing geometrically. if the third arc is20 feet long and the seventh arc is 12 ft. long, what is the length of the arc on the sixth swing? answer is 13.6 feet what are the steps to get this answer?

Answer 1

If the length of the arc traced by such a knot at the bottom of a swinging rope follows a geometric progression, decreasing in this case, the length of each arc will have the following relation:

$A_n=A_{n-1}\times r$

where r stands for the common ratio. We can calculate r and use it to find A6, just as follows:

$\begin{gathered} A_7=A_6\times r=A_5\times r\times r=A_4\times r\times r\times r\ldots=A_3\times r^4 \\ A_7=A_3\times r^4 \end{gathered}$

Substituting the values that were given, we can perform the following calculation:

$\begin{gathered} 12=20\times r^4\to r=\sqrt[4]{\frac{12}{20}} \\ r=\sqrt[4]{0.6} \end{gathered}$

Now we will use it in the following relation:

$\begin{gathered} A_7=A_6\times r\to12=A_6\times\sqrt[4]{0.6} \\ A_6=\frac{12}{\sqrt[4]{0.6}}=\frac{12}{0.8801\ldots}\cong13.6 \\ A_6=13.6ft \end{gathered}$