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Radian Measure

Grade 11 · Trigonometry · Worksheet 3

  1. An astronomer is tracking a comet's path through the solar system. The comet follows a nearly circular orbit with a radius of 3.8 × 10^8 km. During a particular observation window, the comet sweeps out a central angle of 1.9 radians. What distance does the comet travel along its orbital path during this observation?
    Answer: ______________
  2. Mason is examining a circular gear in a machine. The gear has a radius of 9 centimeters. A small mark on the edge of the gear rotates through an angle of 7π/12 radians. What is the exact length of the arc traced by the mark along the circumference of the gear? (Express your answer in terms of π.)
    Answer: ______________
  3. Amelia is a cartographer creating a map of a circular city park. The park has a radius of 20 kilometers. Two landmarks are located along the park's boundary, and the angle between the lines from the center to these landmarks is 5 radians. What is the distance along the curved path between the two landmarks?
    Answer: ______________
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Answer Key & Explanations

Radian Measure · Grade 11 · Worksheet 3

  1. An astronomer is tracking a comet's path through the solar system. The comet follows a nearly circular orbit with a radius of 3.8 × 10^8 km. During a particular observation window, the comet sweeps out a central angle of 1.9 radians. What distance does the comet travel along its orbital path during this observation? Answer: 7.22 × 10^8 km 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: Identify the given values: r = 3.8 × 10^8 km, θ = 1.9 radians Step 3: Substitute the values into the formula: s = (3.8 × 10^8) × 1.9 Step 4: Multiply the numbers: 3.8 × 1.9 = 7.22 Step 5: Apply the exponent: 7.22 × 10^8 km Step 6: The comet travels 7.22 × 10^8 km along its orbital path. The answer is 7.22 × 10^8 km.

  2. Mason is examining a circular gear in a machine. The gear has a radius of 9 centimeters. A small mark on the edge of the gear rotates through an angle of 7π/12 radians. What is the exact length of the arc traced by the mark along the circumference of the gear? (Express your answer in terms of π.) Answer: 21π/4 centimeters Solution: The arc length formula for a circle is s = r * θ, where r is the radius and θ is the central angle in radians. Substitute the given values: r = 9 cm, θ = 7π/12 radians. s = 9 * (7π/12) = (9 * 7π)/12 = 63π/12.
    Full step-by-step solution

    Step 1: The arc length formula for a circle is s = r * θ, where r is the radius and θ is the central angle in radians. Step 2: Substitute the given values: r = 9 cm, θ = 7π/12 radians. Step 3: s = 9 * (7π/12) = (9 * 7π)/12 = 63π/12. Step 4: Simplify the fraction: 63/12 = 21/4. Step 5: Therefore, s = 21π/4 cm. The exact arc length is 21π/4 centimeters.

  3. Amelia is a cartographer creating a map of a circular city park. The park has a radius of 20 kilometers. Two landmarks are located along the park's boundary, and the angle between the lines from the center to these landmarks is 5 radians. What is the distance along the curved path between the two landmarks? Answer: 100 Solution: The distance along a circular arc is given by s = rθ, where r is the radius and θ is the central angle in radians. Here, r = 20 km and θ = 5 radians. s = 20 × 5 = 100.
    Full step-by-step solution

    Step 1: The distance along a circular arc is given by s = rθ, where r is the radius and θ is the central angle in radians. Step 2: Here, r = 20 km and θ = 5 radians. Step 3: s = 20 × 5 = 100. The distance along the curved path between the two landmarks is 100 kilometers.