Circles

[Davekim]

I'm having trouble with this question,

"A circle with a circumference of 6π ft. has a central angle that traces out an arc length of 4 ft. What is the radian measure of the central angle?"

Can someone please explain the process on how to do this problem so I can understand how to do it?

 

[MarkFL]

Hello, and welcome to FMH!

The arc-length \(s\), radius \(r\) and central angle \(\theta\) are related as follows:

\(\displaystyle s=r\theta\tag{1}\)

The circumference \(C\) and radius \(r\) of a circle are related as follows:

\(\displaystyle C=2\pi r\tag{2}\)

What I would do is solve (1) for \(\theta\), and then solve (2) for \(r\), and substitute for \(r\) into the equation resulting from solving (1) for \(\theta\). Then you will have \(\theta\) as a function of \(s\) and \(C\), both of which you've been given. What do you get?

 

[MarkFL]

To follow up:

Solving (1) for \(\theta\) we obtain:

\(\displaystyle \theta=\frac{s}{r}\)

Solving (2) for \(r\) we obtain:

\(\displaystyle r=\frac{C}{2\pi}\)

Hence:

\(\displaystyle \theta=\frac{s}{\dfrac{C}{2\pi}}=\frac{2\pi s}{C}\)

Plugging in the given data:

\(\displaystyle \theta=\frac{2\pi(4\text{ ft})}{6\pi\text{ ft}}=\frac{4}{3}\)