Final answer:
The areas of the sectors formed by ∠UTV can be found using the formula for the area of a sector. For the given central angles, the correct areas are approximately 10.5 and 18.5. Therefore, option (a) is the correct answer.
Step-by-step explanation:
The areas of the sectors formed by ∠UTV can be found using the formula for the area of a sector, which is A = (θ/360) * π * r^2, where θ is the measure of the central angle and r is the radius of the circle. In this case, the measures of the central angles are given: (a) 30.1° and (b) 48.7°.
(a) The area of the sector with a central angle of 30.1° is A = (30.1/360) * π * r^2. You are given two possible answers: 10.5 and 31.4. We can determine which one is correct by substituting each answer and solving for r. If 10.5 is the correct answer, then 10.5 = (30.1/360) * π * r^2. Solving for r gives us r = sqrt((10.5 * 360) / (30.1 * π)). Evaluating this expression gives us r ≈ 5.75. We can then substitute this value of r into the formula to find the area: A ≈ (30.1/360) * π * (5.75^2) ≈ 10.5. Therefore, the correct answer is 10.5.
(b) The process for finding the correct answer for the sector with a central angle of 48.7° is the same. Substituting each possible answer into the formula and solving for r, we find that the correct answer is 18.5. Therefore, the areas of the sectors formed by ∠UTV are approximately 10.5 and 18.5, which matches option (a).