If uc=5, and the area of the sector of the circle enclosed by the central angle subtended by un⌢ is 6.54 in2, what is m∠ucn? round to nearest degree
a: 30
b: 150
c: 27
d: 330

m∠UCN = 30° to the nearest degree ⇒ answer (A)

Step-by-step explanation:

* Lets talk about the area of the sector

- Any sector of a circle formed by 2 radii and an arc

- The central angle formed from the two radii is subtended by this arc

- The area of this sector = (1/2) r²Ф, where r is the length of the radius

  and Ф is the central angle with radiant measure

* Lets revise how to change the measure of the angle from radiant to

 degree or from degree to radiant

⇒Ф° × π/180 = Ф radiant

⇒Ф radiant × 180/π = Ф degree

* Now lets solve the problem

- UC is the radius of the circle C

- Un is an arc subtended to the central angle UCN

∵ UC = 5 ⇒ r = 5 inches

∵ The area of the sector enclosed by the central angle UCN (Ф)

  and the two radii CU and CN is 6.54 inches²

∵ Area the sector = (1/2)r²Ф

∴ (1/2)(5)²Ф = 6.54

∴ 25/2 Ф = 6.54 ⇒ divide the both sides by 25/2

∴ Ф = 6.54/(25/2) = 0.5232 radiant

* Lets change the measure from radiant to degree

∴ Ф° = 0.5232 × (180/π) = 29.977 ≅ 30° to the nearest degree

* The measure of the central angle UCN is 30°