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

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: 8 feet
Central Angle, θ: θ = 225°

Accepted Answer

Hello, today we're going to be solving the following arc length problem. So we are told to calculate the arc length of a circle with a radius of 14 ft. The circle has a central angle of 310 degrees. We want to write our answer in terms of pi and in a decimal form, rounded to the nearest hundreds. So let's go ahead and draw an image to help us understand what is going on in the problem. We have this sector of a circle. We are told that the circle has a radius of 14 ft. So we have R is equal to 14. We are also told that the central angle of the circle is 310 degrees. So the angle inside the sector is which is going to be theta. We want to go ahead and calculate the length of the arc that is formed by this sector. So how do we go about doing that? Well, we have a formula that helps us calculate the arc length of a sector and that is given to us as S is equal to R times data. In this case. S represents the arc length R represents the radius of the circle and theta represents the central angle. So we need to go ahead and get our R and theta value in order to calculate the arc length. Now, in this case here, we already know what R is R is given to us as 14. We also know the value of the, the is equal to 310 degrees. Now keep in mind that the problem wants us to write or answer in terms of pie. So in order to do that, we're going to have to convert data from degrees to radiant. In order to do this, we're going to multiply 310 by the quantity pi over 80 doing so is going to give us the value of 310 pie over 180. And this value is going to simplify to give us pie over 18. So this is going to be the value of the in terms of radiance now that we have our R value and the value we can just go ahead and plug this into the equation for the arc length. So doing so is going to give us S is equal to R which is 14 multiplied by theta which is 31 pi over 18, multiplying these values together is going to give us the value of 434 pi over 18. And this value is going to simplify to give us pi over nine. So this is going to be the arc length in terms of pie. But keep in mind that the problem asks us to also give the answer in the form of a decimal, rounded to the nearest 100th. In order to do this, you can plug in this value into a calculator and round the following decimal to the nearest 100th place. And doing so should give you an approximate decimal of 75.75. So this is going to be the arc length in the form of a decimal. So just to summarize, we use the formula for arc length to calculate the arc length in terms of pi and in the form of a decimal. And with that being said, the answer to this problem is going to be a. 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: 8 feet Central Angle, θ: θ = 225°

Key Concepts

Arc Length Formula

Conversion from Degrees to Radians

Expressing in Terms of π

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