Work each problem. See Example 5. ...

Question

Work each problem. See Example 5.                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                                                                      A​​​​ngle Measure Find t​​he measure (in radians) of a central angle of a sector of area 16 in² a circle of radius 3.0 in.

Accepted Answer

Welcome back. I am so glad you're here. We're told the radius R and area A of a sector of a circle are given below. Calculate the central angle in radiance. Write your answer to one decimal place. We're told that the area equals 27 ft squared and the radius equals 5.0 ft. Our answer choices are answer choice A 2.2 radiance answer choice B 1.2 radiance answer choice C 5.2 radiance and answer choice D 4.8 radiance. All right, we recall from previous lessons that the formula for the area of a sector of a circle is A equals one half R squared beta. And we know some of this, we know the A, the A is 27 ft squared and we can bring down equals one half multiplied by. And we've got R R is 5.0 ft, but that needs to be squared and then that's multiplied by theta. Well, the only unknown variable here is theta and that is what we're trying to find calculating the central angle in radiance. So really, now it's just mathematically solving this for theta. So we can do some simplification. First, if we take the 5.0 ft squared, that's going to be the five squared is 25 and the feet squared is feet squared. And then for that multiplied by one half, what we could do is cancel that out first multiplying both sides by two to get rid of that one half on the right hand side. And then on the left, 27 ft squared equals ft squared. So we have 54 ft squared equals 25 ft squared multiplied by theta. And now we can divide both sides by 25 ft squared. And the feet squared on the left, cancel out the 25 ft squared on the right cancels out. And on the right, we just have theta and that equals on the left 54 divided by 25 which is two point two. When we are rounding to one decimal place, you might get on your calculator, 2.16. So 2.2 equals theta and that is in radiance. So we can say that our theta equals 2.2 radiance and we look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

Work each problem. See Example 5. Angle Measure Find the measure (in radians) of a central angle of a sector of area 16 in² a circle of radius 3.0 in.

Key Concepts

Central Angle

Area of a Sector

Radians

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