In Exercises 71–74, find the length of the arc on a circle of ...

Question

In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 12 inches Central Angle, θ: θ = 45°

Accepted Answer

Hello, today we're gonna be solving the following arc length problem. So what we are told is that we want to calculate the arc length of a circle with a radius of 18 inches. The arc length is intercepted by a central angle of 32 degrees. We want to write the answer in terms of pi and in a decimal form rounded to the nearest 100th place. So let's go ahead and draw an image to help us understand what's going on. We have the circle and the circle has a sector inside of it. We are told that the radius of the circle is 18 inches. So R is equal to 18. We are also told that the central angle of the circle is going to be degrees. So theta is equal to 32 if you want to use this information to calculate the length of the arc that is formed with these values. So how do we go about calculating the arc length? Well, we have a formula that helps us calculate the arc length of a sector of a circle. That formula is given to us as S is equal to R times theta here R represents the radius of the circle and theta represents the central angle. And we already have those values labeled by the circle on the left R is going to equal to 18 and theta is equal to 32. Now, one thing to note is that the answer or the problem asks us to describe the answer in terms of pi, in order to do this, we need to convert data from degrees to radiance. In order to do this, we're gonna multiply 32 by the quantity pi over 180. Doing so is going to give us 32 pie over 180. And this value is going to simplify to give us pie over 45. So this is going to be the form of data in the form of a radiant and now that we have our R value and our, the value we can plug in these two values into the arc length in order to solve this problem. Doing so is going to give us the equation S is equal to R which is 18 multiplied by theta which is eight pi over 45 multiplying these values together is going to give us 144 pi over 45. And this value is going to simplify to give a 16 pie over five. So this is going to be the value of the arc length in terms of pi. But keep in mind that we also want the answer to be in the form of a decimal rounded to the nearest 100th. In order to do this, you can plug in 16 pi over over five into a calculator and round the resulting decimal to the nearest 100th place. And doing so should give you an approximate decimal of 10.5. So just to summarize, we use the equation of the arc length to calculate the arc length in the form of terms of pi and in a decimal rounded to the nearest 100th. And with that being said, the answer to this problem is going to be c so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 12 inches Central Angle, θ: θ = 45°

Key Concepts

Arc Length Formula

Conversion Between Degrees and Radians

Expressing Answers in Terms of π and Decimal Approximation

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