Work each problem. See Example 5. ...

Question

Work each problem. See Example 5.                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                      Arc Length A circular sector has an area of  50 in² . The radius of the circle is 5 in. What is the arc length of the sector?

Accepted Answer

Welcome back. Everyone in this problem, the radius R and area of a sector are given as area equals 64 square feet and the radius equals 8 ft. And we want to calculate the arc length of our sector. For our answer choices, A is 16 ft. B is 32 ft, C is 64 ft and the D is 8 ft. Let's visualize this to make sense of what's going on here. So we're talking about a sector, let me draw our sector on the right. And for our sector, we don't know what angle the sector is what we know that its area is 64 square feet. OK? So let me imagine that we're shading our area here. And we also know that its radius is ft. So I can draw my dimension for the radius here on the left of the sector. So R equals 8 ft. What else do we know? We know that we're trying to calculate the arc length of the sector. In other words, we're trying to calculate that area highlighted in yellow. And let's call that, that, that, that section s that length. S no, we already know that for the arc length recall that the arc length equals are multiplied by theta where theta can be in radiant. So if we can solve for theater, given that we already have R, that means we'll be able to solve for the A. Now, what else do we know? Well, since we are given the area we can solve for theater using the formula for the area of a sector of a circle. So solving for the first recall that the area of a sector equals a half multiplied by R squared theta, we know the area we know the radius. So by substituting those values, that means 64 equals a half multiplied by the radius eight squared multiplied by theta. Now eight squared equals 64. So we're multiplying a half by 64 which equals 32. So 64 equals 32 theta. That means then that theta would be equal to divided by 32 which equals two radiant. So theta is two radiant know that we have a value for a theater we will be able to solve for the A length. So solving, let me come over to the right under my diagram, solving for our arc length S that means our arc length would be equal to R which is it multiplied by theta, which is two radiant and that equals 16. Therefore, our arc length is 16 ft. When we look back on our answer choices, that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

Work each problem. See Example 5. Arc Length A circular sector has an area of 50 in² . The radius of the circle is 5 in. What is the arc length of the sector?

Key Concepts

Area of a Circular Sector

Arc Length Formula

Conversion Between Area and Angle

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