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Radian Measure

Grade 11 · Trigonometry · Worksheet 1

  1. An astronomer is tracking the International Space Station as it orbits Earth. During a 30-minute observation window, the ISS sweeps out a central angle of 1.8 radians relative to Earth's center. If the ISS orbits at an altitude where its orbital radius (distance from Earth's center) is 6,780 km, what distance along its orbital path does the ISS travel during this observation period? Answer: ______________
  2. A circle has radius 8 cm. An arc subtends an angle of 2π/3 radians at the center. Find the arc length.
    Answer: ______________
  3. A robotic arm in a manufacturing plant rotates through a central angle of 3π/4 radians to move components between two workstations. If the tip of the arm travels an arc length of 4.71 meters during this rotation, what is the length of the robotic arm from its pivot point to the tip? Answer: ______________
  4. A circular sector has a central angle of 2π/3 radians and an arc length of 8π cm. A second circular sector has the same radius but a central angle of π/4 radians. What is the arc length of the second sector? Answer: ______________
  5. A central angle of π/3 radians intercepts an arc of length 4π cm. Find the radius of the circle. Answer: ______________
  6. An engineer is designing a curved highway exit ramp that forms a circular arc. The ramp subtends a central angle of 2.8 radians and has a total length of 210 meters. What is the radius of the circular curve that defines this highway ramp? Answer: ______________
  7. A circular water sprinkler rotates back and forth through a central angle of 3π/4 radians. The sprinkler's spray reaches exactly 7 meters from the pivot point. What is the exact length of the arc (in meters) that the sprinkler covers during one complete back-and-forth cycle? Answer: ______________
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Answer Key & Explanations

Radian Measure · Grade 11 · Worksheet 1

  1. An astronomer is tracking the International Space Station as it orbits Earth. During a 30-minute observation window, the ISS sweeps out a central angle of 1.8 radians relative to Earth's center. If the ISS orbits at an altitude where its orbital radius (distance from Earth's center) is 6,780 km, what distance along its orbital path does the ISS travel during this observation period? Answer: 12204 km Solution: Recall the formula for arc length: s = r × θ, where s is arc length, r is radius, and θ is the central angle in radians. Identify the given values: r = 6,780 km and θ = 1.8 radians.
    Full step-by-step solution

    Step 1: Recall the formula for arc length: s = r × θ, where s is arc length, r is radius, and θ is the central angle in radians. Step 2: Identify the given values: r = 6,780 km and θ = 1.8 radians. Step 3: Apply the formula: s = 6,780 × 1.8 Step 4: Calculate: 6,780 × 1.8 = 12,204 Step 5: Include units: The distance traveled is 12,204 km. The ISS travels 12,204 km during the 30-minute observation period.

  2. A circle has radius 8 cm. An arc subtends an angle of 2π/3 radians at the center. Find the arc length. Answer: 16π/3 Solution: Arc length = radius × angle in radians Identify the given values. Radius, r = 8 cm Angle, θ = 2π/3 radians Substitute the values into the formula. Arc length = 8 × (2π/3) Multiply the numbers.
    Full step-by-step solution

    To find the arc length, we use the formula: Arc length = radius × angle in radians Step 1: Identify the given values. Radius, r = 8 cm Angle, θ = 2π/3 radians Step 2: Substitute the values into the formula. Arc length = 8 × (2π/3) Step 3: Multiply the numbers. 8 × (2π/3) = (8 × 2π) / 3 = 16π / 3 Step 4: State the final answer with units. The arc length is 16π/3 cm. Therefore, the final answer is 16π/3.

  3. A robotic arm in a manufacturing plant rotates through a central angle of 3π/4 radians to move components between two workstations. If the tip of the arm travels an arc length of 4.71 meters during this rotation, what is the length of the robotic arm from its pivot point to the tip? Answer: 2 meters 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. Identify the given values: s = 4.71 meters, θ = 3π/4 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: Identify the given values: s = 4.71 meters, θ = 3π/4 radians. Step 3: Substitute the known values into the formula: 4.71 = r × (3π/4) Step 4: Calculate 3π/4: 3 × 3.14159 ÷ 4 = 9.42477 ÷ 4 = 2.35619 Step 5: Solve for r: r = 4.71 ÷ 2.35619 = 2 Step 6: The length of the robotic arm is 2 meters.

  4. A circular sector has a central angle of 2π/3 radians and an arc length of 8π cm. A second circular sector has the same radius but a central angle of π/4 radians. What is the arc length of the second sector? Answer: 3π cm Solution: arc length = radius × central angle (in radians) Central angle = 2π/3 radians Arc length = 8π cm 8π = r × (2π/3) 8π = r × (2π/3) 8 = r × (2/3) Multiply both sides by 3/2: r = 8 × (3/2) = 12 cm So the radius is 12 cm.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Understand the relationship for the first sector** A circular sector has: arc length = radius × central angle (in radians) For the first sector: Central angle = 2π/3 radians Arc length = 8π cm So: 8π = r × (2π/3) --- **Step 2: Solve for the radius r** 8π = r × (2π/3) Divide both sides by π: 8 = r × (2/3) Multiply both sides by 3/2: r = 8 × (3/2) = 12 cm So the radius is 12 cm. --- **Step 3: Apply to the second sector** Second sector: Same radius r = 12 cm Central angle = π/4 radians Arc length = r × central angle Arc length = 12 × (π/4) --- **Step 4: Simplify** 12 × (π/4) = (12/4) × π = 3 × π = 3π cm --- **Final Answer:** 3π cm

  5. A central angle of π/3 radians intercepts an arc of length 4π cm. Find the radius of the circle. Answer: 12 Solution: Central angle = π/3 radians Arc length = 4π cm We need the radius r. Recall the formula for arc length. Arc length = radius × central angle in radians s = r × θ Substitute the known values into the formula.
    Full step-by-step solution

    We are given: Central angle = π/3 radians Arc length = 4π cm We need the radius r. Step 1: Recall the formula for arc length. Arc length = radius × central angle in radians So: s = r × θ Step 2: Substitute the known values into the formula. s = 4π θ = π/3 So: 4π = r × (π/3) Step 3: Solve for r. Divide both sides by π: 4 = r × (1/3) Multiply both sides by 3: r = 4 × 3 r = 12 Step 4: State the answer. The radius of the circle is 12 cm.

  6. An engineer is designing a curved highway exit ramp that forms a circular arc. The ramp subtends a central angle of 2.8 radians and has a total length of 210 meters. What is the radius of the circular curve that defines this highway ramp? Answer: 75 meters 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 known values: 210 = r × 2.8 Step 3: Solve for r by dividing both sides by 2.8: r = 210 ÷ 2.8 Step 4: Calculate: 210 ÷ 2.8 = 75 Step 5: The radius is 75 meters.

  7. A circular water sprinkler rotates back and forth through a central angle of 3π/4 radians. The sprinkler's spray reaches exactly 7 meters from the pivot point. What is the exact length of the arc (in meters) that the sprinkler covers during one complete back-and-forth cycle? Answer: 21π/2 Solution: The sprinkler rotates through 3π/4 radians in one direction, then returns through the same angle. The total angle for one complete back-and-forth cycle is 3π/4 + 3π/4 = 6π/4 = 3π/2 radians.
    Full step-by-step solution

    Step 1: The sprinkler rotates through 3π/4 radians in one direction, then returns through the same angle. The total angle for one complete back-and-forth cycle is 3π/4 + 3π/4 = 6π/4 = 3π/2 radians. Step 2: The arc length formula is s = rθ, where r is the radius and θ is the angle in radians. Step 3: Substitute the known values: s = 7 × (3π/2) Step 4: Calculate: s = 21π/2 Step 5: The exact arc length is 21π/2 meters.