CONCEPT PREVIEW Find the area of each sector.

Question

Accepted Answer

Hello, everyone. We are asked to determine the area of the indicated sector of the following circle. We're giving a circle where the marked arc length is 48 pi we have theta marked, but we do not have a value for the angle theta and we have a radius of 32. Our answer choices are a two pi divided by three square units. B three pi divided by two square units, C 24 pi square units, D 768 pi square units. So the first thing I want to find is the, so to find theta, I'm gonna use the formula where we know S equals R multiplied by theta. So in this S is going to be our arc length. And in our problem that is 48 pi and R is our radius and our radius here is 32. So putting those values into the equation, I get 48 pi equals 32 multiplied by the angle theta dividing both sides by 32 to find the value of theta. And after I reduce, I get that theta equals three pi divided by two. So we're on the right track, we know theta is three pi divided by two. We need to find the area of that sector though. So the formula for area of a sector is that a equals one half multiplied by the radius squared multiplied by theta. So here we'll have a equals one half multiplied by our radius which is 32 squared multiplied by theta, which is three pi divided by two. I'm gonna find out what 32 squared is first. So I have one half multiplied by 1024 multiplied by three pi divided by two. And now I'm gonna multiply straight across my numerators and then my denominator. So straight across the numerator, I will get 3072 pie and across the denominator, I get four. Now I want to simplify that and see if 3072 can be divided by four and it can, it comes out to 768 we still have pie because there's nothing to cancel out pie. And since we don't have a specific unit, this is why it is called square units. So our area here is 768 pie square units which is answer choice. D have a nice day.

CONCEPT PREVIEW Find the area of each sector.

Key Concepts

Definition of a Sector

Formula for Area of a Sector

Conversion Between Radians and Degrees

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