Radian Measure

lessonbunny.com

Answer Key & Explanations

Radian Measure · Grade 11 · Worksheet 3

1. A sector of a circle has area 24π cm² and central angle 2π/3 radians. Find the radius of the circle: r = ? Answer: 6 cm Solution: The formula for the area of a circular sector is A = (1/2) × r² × θ, where A is the area, r is the radius, and θ is the central angle in radians.
   Full step-by-step solution

   Step 1: The formula for the area of a circular sector is A = (1/2) × r² × θ, where A is the area, r is the radius, and θ is the central angle in radians. Step 2: Substitute the given values: 24π = (1/2) × r² × (2π/3) Step 3: Simplify the right side: (1/2) × (2π/3) = π/3, so 24π = (π/3) × r² Step 4: Divide both sides by π: 24 = (1/3) × r² Step 5: Multiply both sides by 3: 72 = r² Step 6: Take the square root of both sides: r = √72 = √(36×2) = 6√2 Step 7: Since the problem asks for the radius and the area was given in cm², the radius is 6√2 cm. The answer is 6√2 cm.

2. A sector of a circle has radius 12 cm and arc length 10π cm. Find the central angle in radians: θ = ? Answer: 5π/6 Solution: Use the arc length formula: s = rθ, where s is arc length, r is radius, and θ is central angle in radians.
   Full step-by-step solution

   Step 1: Use the arc length formula: s = rθ, where s is arc length, r is radius, and θ is central angle in radians. Step 2: Substitute the given values: 10π = 12 × θ Step 3: Solve for θ: θ = (10π)/12 Step 4: Simplify the fraction: θ = (5π)/6 Step 5: The central angle is 5π/6 radians.

3. A central angle of π/3 radians intercepts an arc of length 5π cm. Find the radius of the circle. Answer: 15 Solution: Central angle = π/3 radians Arc length = 5π cm We need the radius r. Recall the formula for arc length.
   Full step-by-step solution

   We are given: Central angle = π/3 radians Arc length = 5π cm We need the radius r. Step 1: Recall the formula for arc length. Arc length (s) = radius (r) × central angle in radians (θ) So, s = r × θ Step 2: Substitute the known values into the formula. 5π = r × (π/3) Step 3: Solve for r. Divide both sides by π: 5 = r × (1/3) Multiply both sides by 3: r = 15 Step 4: Conclusion. The radius of the circle is 15 cm.

4. A circular race track has a radius of 150 meters. A car travels along the track covering a central angle of 2.4 radians. What is the distance traveled by the car along the track? Answer: 360 meters Solution: We have a circular track with radius r = 150 meters. The car travels along the track covering a central angle θ = 2.4 radians. We need the arc length (distance traveled along the track).
   Full step-by-step solution

   Step 1: Understand the problem We have a circular track with radius r = 150 meters. The car travels along the track covering a central angle θ = 2.4 radians. We need the arc length (distance traveled along the track). Step 2: Recall the formula for arc length For a circle, the arc length s is given by: s = r × θ where r is the radius and θ is the central angle in radians. Step 3: Substitute the given values r = 150 m θ = 2.4 radians So: s = 150 × 2.4 Step 4: Perform the multiplication 150 × 2.4 = 150 × (24/10) = (150 × 24) / 10 150 × 24 = 3600 3600 / 10 = 360 Step 5: State the final answer with units The distance traveled by the car along the track is 360 meters.

5. Liam is designing a circular running track for his school's new athletic facility. The track has a radius of 45 meters and the architect needs to install special rubberized surfacing along an arc that subtends an angle of 2.1 radians at the center. Calculate the exact length of this arc that requires the special surfacing. Answer: 94.5 Solution: We have a circular track with radius r = 45 meters. We need the length of an arc that subtends an angle θ = 2.1 radians at the center. s = r × θ where r is the radius and θ is the angle in radians.
   Full step-by-step solution

   Step 1: Understand the problem We have a circular track with radius r = 45 meters. We need the length of an arc that subtends an angle θ = 2.1 radians at the center. Step 2: Recall the formula for arc length For a circle, the arc length s is given by: s = r × θ where r is the radius and θ is the angle in radians. Step 3: Substitute the given values r = 45 m θ = 2.1 radians So: s = 45 × 2.1 Step 4: Perform the multiplication First, 45 × 2 = 90 Then, 45 × 0.1 = 4.5 Add them: 90 + 4.5 = 94.5 Step 5: State the final answer The exact length of the arc is 94.5 meters. Step 6: Reasoning check Since the angle is given in radians, the formula s = r × θ applies directly without needing to convert degrees to radians. The multiplication gives the exact arc length.