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