Concept Check If the radius of a circle is doubled, how is the...

Question

Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

Accepted Answer

Hello, everyone. We are asked to explain briefly what would happen to the arc length intercepted by a stationary central angle. If the radius of the circle is tripled, our answer choices are a, the arc length would decrease by a factor of nine B. The arc length would increase by a factor of nine. See, the arc length would decrease by a factor of three D. The arc length would be increased by a factor of three. First recall that we have a formula S which is our arc length equals R as a radius multiplied by the central angle theta. We're gonna set this equation up twice. So we have our original circle with its original ratio, our radius. So I'm gonna call that S sub one equals R sub one multiplied by theta sub one. And then we have our newer circle and I'm going to say that's S sub two equals R sub two multiplied by theta sub two. So in this, we are told that in our newer circle, our radius are sub two is equal to three times the original radius of R sub one. So I can say R sub two equals three R sub one. So I'm gonna substitute that into the formula. So I've s sub two equals three are sub one theta sub two. Now that I have two equations, I'm gonna set them as a ratio of our newer circle over our older circle. So I'm gonna set up the ratio S sub two divided by S sub one. And that's gonna equal three are sub one theta sub two divided by R sub one, theta sub one. So I have my new circles equation divided by my old circles equation. Now that I have this set up, I'm going to try to reduce what I can. So our sub ones will cancel. We were told in the problem that we have the same stationary central angle. So that means that theta sub two equals theta sub one. So we can reduce those also. And when we do so the ratio of the arc length is going to be three times greater. So this means that our arc length, when we do the second circle, the newer circle divided by the first circle would increase by a factor of three. So that's answer choice D the arc length would be increased by a factor of three. Have a nice day.

Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

Key Concepts

Arc Length Formula

Effect of Radius on Arc Length

Central Angle in Radians

Watch next