Radian Measure

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Answer Key & Explanations

Radian Measure · Grade 11 · Worksheet 2

1. Liam is designing a circular garden with a central fountain. The garden has a radius of 8 meters, and he wants to install a decorative stone path along an arc that subtends an angle of 2.1 radians at the center. What is the length of the stone path Liam needs to install? Answer: 16.8 Solution: We have a circular garden with radius r = 8 meters. Liam wants a stone path along an arc of the circle. The arc subtends an angle θ = 2.1 radians at the center.
   Full step-by-step solution

   Step 1: Understand the problem We have a circular garden with radius r = 8 meters. Liam wants a stone path along an arc of the circle. The arc subtends an angle θ = 2.1 radians at the center. We need the arc length. 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 = 8 m θ = 2.1 radians So: s = 8 × 2.1 Step 4: Perform the multiplication 8 × 2.1 = 8 × (2 + 0.1) = 16 + 0.8 = 16.8 Step 5: State the final answer The length of the stone path is 16.8 meters. Step 6: Check the reasoning Since the angle is already in radians, we don’t need to convert units. The formula s = rθ works directly, giving 16.8 m. This matches the correct answer.

2. Zoe is designing a circular fountain for a park. The fountain will have a radius of 12 meters. A decorative light strip will be installed along an arc that subtends a central angle of 1 radians. How long, in meters, will the light strip be? Answer: 12 Solution: Recall the formula for arc length: s = rθ, where r is the radius and θ is the central angle in radians. Substitute the given values: r = 12 meters, θ = 1 radians. Calculate: s = 12 × 1 = 12.
   Full step-by-step solution

   Step 1: Recall the formula for arc length: s = rθ, where r is the radius and θ is the central angle in radians. Step 2: Substitute the given values: r = 12 meters, θ = 1 radians. Step 3: Calculate: s = 12 × 1 = 12. The decorative light strip will be 12 meters long.

3. 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.

4. A sector of a circle has area 24π cm² and arc length 8π cm. Find the radius of the circle. Answer: 6 cm Solution: Write the formulas for sector area and arc length. Sector area A = (1/2)r²θ Arc length s = rθ Substitute the given values. 24π = (1/2)r²θ 8π = rθ Solve the arc length equation for θ.
   Full step-by-step solution

   Step 1: Write the formulas for sector area and arc length. Sector area A = (1/2)r²θ Arc length s = rθ Step 2: Substitute the given values. 24π = (1/2)r²θ 8π = rθ Step 3: Solve the arc length equation for θ. θ = 8π/r Step 4: Substitute this into the area equation. 24π = (1/2)r²(8π/r) Step 5: Simplify the equation. 24π = (1/2)r(8π) 24π = 4πr Step 6: Solve for r. 24π/4π = r 6 = r The radius of the circle is 6 cm.