In Exercises 91–92, find the measure of the central angle on a ci...

Question

In Exercises 91–92, find the measure of the central angle on a circle of radius r that forms a sector with the given area.

Radius, r: 10 feet
Area of the Sector, A: 25 square feet

Accepted Answer

Hello, today we're gonna be solving the following problem. So we are told that a sector of a circle has an area of 48 square feet. The circle's radius is 16 ft. We want to solve for the central angle that forms the sector and express the answer in radiance. So let's go ahead and draw an image for the given information. So we're talking about a sector of a circle or in this case, a wedge of a circle, we are told that this sector has an area of 48 square feet. So we can say a is equal to 48. We are also told that the circle's radius is 16 ft long. Our goal is to solve for the inner angle theta. Then we want to express the answer in terms of radiance. So how do we go about doing this? Well, we're going to have to use the formula for the area of a sector of a circle. The area for the sector of a circle is given to us as a is equal to one half times radius squared times theta. And here our goal is to solve for data. So let's just make a list of a few things that are given to us. We are given the area a which we can substitute into the formula. And we are given the radius which is 16 ft, which we can also substitute into the formula. So if we use this formula with the given information, we can write the formula as 48 is equal to one half times the quantity 16 squared times theta. So let's go ahead and use this to solve for data squared is going to simplify to give us 256 and one half multiplied by 256 is going to give us 128. So we have 48 is equal to multiplied by the. Finally, in order to solve our data, we're going to have to divide both sides of this equation by 128. Doing so is going to give us data is equal to 48 over 128. And this value is going to simplify to give us 3/8. Now keep in mind that data is already in the form of radiance. So we can say that the central angle is 3/8 radiant long. And this is going to be the solution to this problem. And with that being said, the answer to this problem is going to be D. 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.

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In Exercises 91–92, find the measure of the central angle on a circle of radius r that forms a sector with the given area. Radius, r: 10 feet Area of the Sector, A: 25 square feet