Radian Measure

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Answer Key & Explanations

Radian Measure · Grade 11 · Worksheet 2

1. An engineer is designing a curved highway exit ramp that follows the arc of a circle with radius 120 meters. The ramp needs to be exactly 94.2 meters long. What central angle (in radians) should the engineer use for this curved section of the highway? Answer: 0.785 Solution: Recall the formula for arc length: s = rθ, where s is arc length, r is radius, and θ is the central angle in radians.
   Full step-by-step solution

   Step 1: Recall the formula for arc length: s = rθ, where s is arc length, r is radius, and θ is the central angle in radians. Step 2: Substitute the given values: 94.2 = 120 × θ Step 3: Solve for θ by dividing both sides by 120: θ = 94.2 ÷ 120 Step 4: Calculate the division: θ = 0.785 Step 5: The central angle should be 0.785 radians.

2. 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.

3. A circle has radius 8 cm and a central angle of 3π/4 radians. Find the arc length = ? Answer: 6π Solution: Recall the formula for arc length. s = r * θ where r is the radius and θ is the central angle in radians. Identify the given values.
   Full step-by-step solution

   Step 1: Recall the formula for arc length. The arc length (s) of a circle is given by: s = r * θ where r is the radius and θ is the central angle in radians. Step 2: Identify the given values. Radius r = 8 cm Central angle θ = 3π/4 radians Step 3: Substitute the values into the formula. s = 8 * (3π/4) Step 4: Simplify the multiplication. First, multiply 8 by 3π/4: 8 * 3π/4 = (8/4) * 3π = 2 * 3π = 6π Step 5: State the final answer with units. Arc length = 6π cm Thus, the correct answer is 6π.

4. A circular water sprinkler sprays water in a pattern that covers a 135° sector of a circle with radius 12 meters. What is the exact length of the outer edge of the watered area? Answer: 9π meters Solution: Convert the central angle from degrees to radians. Since 180° = π radians, 135° = (135/180)π = (3/4)π radians. Use the arc length formula: arc length = radius × angle in radians.
   Full step-by-step solution

   Step 1: Convert the central angle from degrees to radians. Since 180° = π radians, 135° = (135/180)π = (3/4)π radians. Step 2: Use the arc length formula: arc length = radius × angle in radians. Step 3: Substitute the values: arc length = 12 × (3/4)π = (36/4)π = 9π. Step 4: The exact arc length is 9π meters.