Circles

Davekim

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Joined
Apr 11, 2019
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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?
 
Hello, and welcome to FMH! :)

The arc-length ss, radius rr and central angle θ\theta are related as follows:

[MATH]s=r\theta\tag{1}[/MATH]
The circumference CC and radius rr of a circle are related as follows:

[MATH]C=2\pi r\tag{2}[/MATH]
What I would do is solve (1) for θ\theta, and then solve (2) for rr, and substitute for rr into the equation resulting from solving (1) for θ\theta. Then you will have θ\theta as a function of ss and CC, both of which you've been given. What do you get?
 
To follow up:

Solving (1) for θ\theta we obtain:

[MATH]\theta=\frac{s}{r}[/MATH]
Solving (2) for rr we obtain:

[MATH]r=\frac{C}{2\pi}[/MATH]
Hence:

[MATH]\theta=\frac{s}{\dfrac{C}{2\pi}}=\frac{2\pi s}{C}[/MATH]
Plugging in the given data:

[MATH]\theta=\frac{2\pi(4\text{ ft})}{6\pi\text{ ft}}=\frac{4}{3}[/MATH]
 
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