4.4.2: Homework

Math 373: Trigonometry for Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Reading Questions

    1. What is a sector of a circle?
    2. What is the formula for the area of a sector, and what unit must the central angle be in for the formula to be valid?
    3. What is the difference between linear velocity and angular velocity?
    4. What is the formula that relates linear velocity (\(v\)) and angular velocity (\(\omega\)) for an object moving in a circle of radius \(r\)?
    5. What are the typical units for angular velocity?
    6. What is the difference between ordinary frequency and angular frequency?
    7. What is the period of circular motion, and how does it relate to ordinary frequency?
    8. If a point on a circle rotates through an angle of 38° in 3 hours, what must you do first before calculating its angular velocity?
    9. Why is it acceptable to say that the units of \(r\omega\) are \(\frac{\text{length}}{\text{time}}\) even though the units of \(\omega\) are \(\frac{\text{radians}}{\text{time}}\)?

    Homework

    Vocabulary Check

    1. The region bounded by the initial and terminal sides of a central angle \( \theta \) and the edge of the circle is called a ___ of a circle.

    2. A word commonly used in place of spatial position is ___.

    3. The ___ of a particle is the average rate of change of its position with respect to a change in time. If we could take an instantaneous "snapshot" of this form of motion, we would call it the ___ of the particle.

    4. The ___ of a particle is the magnitude of its velocity.

    Concept Check

    1. Write each statement as a mathematical formula.

      1. The length of the arc, \(s\), is equal to:\[\text{(the fraction of a revolution)} \times \text{(circumference of the circle)} \nonumber \]

      2. The area of the sector, \(A\), is equal to:\[\text{(the fraction of a revolution)} \times \text{(area of the circle)}\nonumber \]

    True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.

    1. To compute the area of a sector, we multiply the given angle by half of the square of the circle's radius.

    2. Given two circles, one of radius \( r_1 \) and the other of radius \( r_2 \), particles on each circle moving at a constant angular velocity, \( \omega \), move at the same linear velocity.

    Basic Skills

    For the following exercises, \( \theta \) is the central angle in a circle of radius \( r \). In each case, find the area of the sector.

    1. \( \theta = \frac{2 \pi}{3}, \qquad r = 7 \, \text{yards} \)

    2. \( \theta = \frac{7 \pi}{6}, \qquad r = \sqrt{2} \, \text{m} \)

    3. \( \theta = \frac{6 \pi}{7}, \qquad r = \sqrt{3} \, \text{miles} \)

    4. \( \theta = 3, \qquad r = \frac{1}{2} \)

    5. \( \theta = 5, \qquad r = 4.1 \, \text{cm} \)

    6. \( \theta = 60^{ \circ }, \qquad r = 2 \, \text{feet}\)

    7. \( \theta = 330^{ \circ }, \qquad r = \pi \, \text{inches} \)

    8. \( \theta = 32^{ \circ }, \qquad r = 8 \, \text{parsecs} \)

    9. (MyOpenMath) \( \theta = 212^{ \circ }, \qquad r = 0.88 \, \text{leagues} \)

    For the following exercises, find the area of the shaded sector.

    1.  
      1.  
        Screen Shot 2023-01-06 at 12.23.20 AM.png
      2.  
        Screen Shot 2023-01-06 at 12.23.28 AM.png

    Determining Areas of Sectors

    1. An arc of length 8 feet is cut off by a central angle of \( 2\pi/3 \) radians. Find the area of the resulting sector.

    2. A central angle of 4 radians cuts off an arc of length 16 meters. Find the area of the resulting sector.

    3. (MyOpenMath) Calculate the area of a sector with a radius of 6 cm and a central angle of \( 30^{ \circ } \).

    4. A sector has a radius of 4 meters and a central angle of \( 45^{ \circ } \). Find its area.

    5. (MyOpenMath) If the radius of a sector is 7.4 inches and the central angle is \( 225^{ \circ } \), what is the area of the sector?

    6. Determine the area of a sector with a radius of 8 feet and central angle \( 2\pi/3 \).

    7. Calculate the area of a sector with a 3-meter radius and central angle \( 5\pi/6 \).

    Determining the Radius of a Sector

    1. If the area of a sector is 25 cm2 and the central angle is \( 45^{ \circ } \), find the radius.

    2. Determine the radius of a sector with an area of 30 m2 and a central angle of 60 degrees.

    3. A sector has an area of 14 in² and central angle \( \pi/6 \). What is its radius?

    4. What is the radius of a sector with an area of 78 cm² and a central angle of 120 degrees?

    5. What is the radius of a sector if the area is 100 cm² and the central angle is 250 degrees?

    Determining the Central Angle of a Sector

    1. Find the radian measure of the central angle of a sector with a radius of 5 cm and an area of 20 cm².

    2. Determine the radian measure of the central angle if a sector has a radius of 3 meters and an area of 15 m².

    3. What is the degree measure of the central angle of a sector with a radius of 7 inches and an area of 24 in²?

    4. Calculate the degree measure of the central angle for a sector with a radius of 10 feet and an area of 50 ft².

    5. Calculate the degree measure of the central angle for a sector with a radius of 11 cm and an area of 30.25 cm².

    Determining Angular Velocity

    1. Wheel Rotation. If a wheel rotates 300 times in one minute, find its angular velocity in radians per second.

    2. Merry-go-Round. How many complete revolutions does a merry-go-round make in 10 minutes if it has an angular velocity of 0.5 radians per second?

    3. Mixing Machine. A mixing blade in a blender rotates at 1200 revolutions per minute. Calculate its angular velocity in radians per second.

    4. Computer Hard Drive. The spindle of a computer hard drive spins at 7200 RPM. What is the angular velocity of the spindle in radians per second?

    5. Electric Generator. An electric generator's rotor spins at 3600 revolutions per minute. Determine the rotor's angular velocity in radians per second.

    6. Laundry Machine. During the spin cycle, a washing machine drum rotates at 800 revolutions per minute. Calculate the angular velocity in radians per second.

    7. Vehicle's Alternator. The alternator in a car operates at 6000 RPM when the engine is at full speed. Find the angular velocity of the alternator in radians per second.

    Converting Between Linear and Angular Velocities

    1. Bike Tire. A bicycle tire has a radius of 35 cm. If the tire rotates at an angular velocity of 2 radians per second, calculate the linear velocity of a point on the tire’s rim.

    2. Circular Machinery. A part of a machine moves in a circle of radius 15 cm at a constant speed. If the linear velocity of the part is 90 cm/s, calculate its angular velocity in radians per second.

    3. Ceiling Fan Speed. The blade of a ceiling fan is 0.7 meters long from the center to the tip. If the tip of the blade moves at a linear velocity of 4.5 meters per second, calculate the angular velocity of the blade in radians per second.

    4. Amusement Park Ride. A merry-go-round has a radius of 5 meters. If a horse on the edge of the ride moves at a linear velocity of 3 meters per second, find the angular velocity of the merry-go-round in radians per second.

    5. Industrial Flywheel. An industrial flywheel has a radius of 2 meters. If it is required to achieve a linear speed of 10 meters per second at its rim, determine the necessary angular velocity in radians per second.

    6. Athletics Track Runner. A runner is jogging along a circular track with a radius of 25 meters, maintaining a constant linear speed of 2.5 meters per second. Calculate the runner’s angular velocity in radians per second.

    7. Spinning Top. A spinning top has a radius of 8 cm. If the linear velocity of a point on its edge is 1.6 meters per second during a spin, what is the angular velocity of the top in radians per second?

    8. Windmill Blades. A windmill's blade has an angular velocity of 0.8 radians per second and each blade is 3 meters long. Calculate the linear velocity at the tip of a blade.

    9. Rotating Sign. A circular sign rotates at an angular velocity of 2 radians per second. If the radius of the sign is 1.5 meters, what is the linear velocity of the edge of the sign?

    10. Gear in a Machine. A gear in a factory machine has a radius of 20 cm and rotates with an angular velocity of 5 radians per second. Determine the linear velocity of a point on the edge of the gear.

    11. Carousel Horse. A carousel horse moves in a circular path with a radius of 4 meters. If the angular velocity of the carousel is 1 radian per second, find the linear velocity of the horse.

    12. Bicycle Pedaling Speed. A bicycle's pedal arm is 17.5 cm long, and the cyclist is pedaling at an angular velocity of 4 radians per second. Calculate the linear velocity at the end of the pedal arm.

    Synthesis Questions

    1. Calculate the radius of a sector whose area is one-fifth of the area of a circle with a radius of 10 cm.

    2. Determine the radius of a sector if its area is equivalent to half the area of a circle with a radius of 14 meters.

    3. Two circular saw blades are spinning, one at 120 RPM and the other at 180 RPM. If both blades have a diameter of 30 cm, which blade has a higher linear velocity at the rim, and by how much?

    Applications

    Sector Area Applications

    1. Land Allocation. A farmer plans to allocate a portion of his circular field for a special crop. The field has a radius of 100 meters. If he uses a 60-degree sector for this crop, calculate the area that will be cultivated.

    2. Clock Face Design. A designer is creating a new clock face where each hour marker is represented by a different colored sector. The clock has a radius of 12 cm. Calculate the area of one hour sector assuming a simple 12-hour clock.

    3. Architectural Design. An architect designs a circular window with decorative sectors. Each sector has a central angle of 45 degrees and the window has a diameter of 2 meters. Calculate the area of one decorative sector.

    4. City Planning. A city planner needs to design a roundabout garden that occupies a 38-degree sector of a circle with a radius of 20 meters. Calculate the area available for planting.

    5. Pizza Slicing. A pizza place offers a unique pizza slice shaped as a sector. The pizza has a diameter of 40 cm. If the central angle for each slice is 20 degrees, find the area of one slice of pizza.

    6. Sports Fields. A sports complex includes a sector-shaped running track portion used for sprinting drills. If the track follows a 40-degree sector of a circle with a radius of 100 meters, calculate the area of this track section.

    7. Public Park Design. A public park is designed with a circular area for a water fountain, surrounded by flower beds laid out in sectors. If the circular area has a radius of 15 meters and each sector for the flower beds is 30 degrees, find the area of one flower bed.

    8. Non-Uniform Rainfall. A circular region with a radius of 20 km receives rainfall. However, the rain is not uniformly distributed. If a 120-degree sector receives twice the amount of rain per square kilometer as the rest of the region, calculate the total rainfall if the sector received 600 mm.

    9. Engineering Design. An engineer is designing a gear in the shape of a sector. The gear must mesh with another gear with an area three times larger but with the same arc length. If the radius of the smaller gear is 15 cm, determine the central angle of the sector for the smaller gear.

    10. Advanced Pizza Slicing. A circular pizza is sliced such that each consecutive slice has a central angle 10 degrees greater than the previous one, starting from 10 degrees. Calculate the area of the 5th slice if the pizza has a radius of 30 cm.

    11. Sector-based Cooling Zones. A circular mechanical part is designed with cooling zones in the form of sectors. Each zone dissipates heat differently. If one zone with a 60-degree angle dissipates heat at 100 watts per square meter and the radius of the part is 30 cm, find the total heat dissipation for this zone.
      (Hint: Heat dissipation is the total number of watts of heat released in a given area.)

    Velocity Applications

    1. Find the angular velocity of the following objects.

      1. The London Eye Ferris wheel, which makes one revolution every 20 minutes.

      2. An old-fashioned long-playing record, which revolved \(33 \frac{1}{3}\) times in 60 seconds.

    2. Conveyor Belt. A conveyor belt moves at a linear velocity of 1.5 meters per second. If the driving drum has a radius of 0.5 meters, calculate the angular velocity of the drum in radians per second.

    3. Bike Wheel Spin. A bicycle wheel has a radius of 0.4 meters. If the wheel is spinning with an angular velocity of 5 radians per second, what is the linear velocity of a point on the rim?

    4. Rollerblade Wheel. A rollerblade wheel has a radius of 40 mm. If a rollerblader is moving at a linear speed of 10 km/h, find the angular velocity of the wheels in radians per second.

    5. Ferris Wheel. A Ferris wheel has a radius of 25 meters and rotates at a linear velocity of 2 meters per second at its rim. Calculate its angular velocity in radians per second.

    6. Tire Rotation (MyOpenMath). A truck with 20-inch-diameter tires is traveling at 50 miles per hour.

      1. Find the angular velocity of the tire in radians per minute.

      2. How many revolutions per minute do the tires make?

    7. Tire Rotation. A car tire has a radius of 35 cm. If the car is traveling at 80 km/h, find the angular velocity of the tire in radians per second.

    8. Potter’s Wheel Speed. If the edge of a potter's wheel moves at a linear velocity of 15 m/s and the wheel has a radius of 1.2 meters, determine the angular velocity in radians per second.

    9. Drill Bit Rotation. A drill bit has a radius of 1 cm. If it needs to achieve a linear velocity of 20 m/s at its tip, calculate the required angular velocity in radians per second.

    10. Record Player. If a point on a record player is moving with a linear velocity of 0.5 m/s and the record has a radius of 15 cm, what is the angular velocity in radians per second?

    11. Electric Fan. An electric fan has blades that are each 0.25 meters long. If a point on the tip of a blade moves with a linear velocity of 3 m/s, what is the angular velocity of the blades in radians per second?

    12. Satellite Dish Rotation. A satellite dish has a radius of 2 meters. If it rotates such that a point on its edge moves at a linear velocity of 0.1 m/s, calculate the angular velocity in radians per second.

    13. Fan Speed. A ceiling fan has blades that are each 0.75 meters long. If the fan spins at 200 revolutions per minute (RPM), calculate the angular velocity in radians per second.

    14. Merry-Go-Round. Children are riding on a merry-go-round that has a radius of 2 meters and makes one complete rotation every 4 seconds. Find the angular velocity in radians per second.

    15. Driving Speed and Wheel Rotation. A car with wheels of radius 30 cm is traveling at a speed of 60 km/h. Calculate the angular velocity of the wheels in radians per second.

    16. Bicycle Dynamics. A bicycle's front wheel has a diameter of 70 cm and rotates at 150 RPM while cruising. Determine the linear velocity of the bicycle.

    17. Angular Velocity of a Clock's Second Hand. Calculate the angular velocity in radians per hour of the second hand of a clock.

    18. Disk Rotation. A disk in a laboratory experiment rotates at a speed of 600 RPM. Calculate its angular velocity in radians per second.

    19. Jogging on a Track. A jogger runs around a circular track with a diameter of 200 meters at a speed of 8 km/h. Compute the angular velocity of the jogger in radians per hour.

    20. Wheelchair Wheel Rotations. A wheelchair has wheels of diameter 60 cm. If a person moves at a speed of 4 km/h, determine the angular velocity of the wheels in radians per second.

    21. Spinner Toy. A spinner toy rotates 4000 times in one minute. Calculate its angular velocity in radians per second.

    22. Record Player. A record player spins a record at 33 revolutions per minute. Find the angular velocity in radians per second and the linear velocity at a point 10 cm from the center.

    23. Dryer. Clothes dryers draw 3.5 times as much power as washing machines, so newer machines have been engineered for greater efficiency. A vigorous spin cycle reduces the time needed for drying, and some front-loading models spin at a rate of 1500 rotations per minute.

      1. If the radius of the drum is 11 inches, how far do your socks travel in one minute?

      2. How fast are your socks traveling during the spin cycle?

    24. Hubble Telescope. The Hubble telescope is in orbit around the earth at an altitude of 600 kilometers, and completes one orbit in 97 minutes.

      1. How far does the telescope travel in one hour? (The radius of the earth is 6400 kilometers.)

      2. What is the speed of the Hubble telescope?

    25. Energy-Producing Windmill. The first large windmill used to generate electricity was built in Cleveland, Ohio in 1888. Its sails were 17 meters in diameter, and moved at 10 rotations per minute. How fast did the ends of the sails travel?

    26. Largest Windmill. The largest windmill operating today has wings 54 meters in length. To be most efficient, the tips of the wings must travel at 50 meters per second. How fast must the wings rotate?

    27. Planetary Motion. Assume a simplified model where a planet moves in a circular orbit around its star with a constant angular velocity. If the radius of the orbit is \( 1.5 \times 10^6 \) kilometers and it completes one orbit in 215 days, find the linear velocity of the planet.

    28. Wheel Rotation. A car's tire has a diameter of 60 cm. If the car is traveling at a speed of 72 km/h, calculate the angular velocity of the tire in radians per second.

    29. CD Player Speed. A CD rotates at 500 revolutions per minute (RPM) when playing. What is the linear velocity at a point 3 cm from the center of the CD?

    30. Angular Speed of a Wind Turbine. A wind turbine blade is 45 meters long, and it completes one rotation every three seconds. Calculate the angular velocity in radians per second and the linear velocity at the tip of the blade.

    31. Bicycle Wheel Dynamics. A bicycle wheel has a radius of 35 cm. If the bicycle is moving at a speed of 15 km/h, find the wheel’s angular velocity in radians per second.

    32. Roller Coaster Speed. A roller coaster moves in a vertical loop of radius 12 meters at a speed of 36 km/h at the highest point of the loop. Calculate the angular velocity at this point.

    33. Flywheel Operation. A flywheel in a machine spins at 1800 RPM. If the radius of the flywheel is 20 cm, calculate the linear velocity at the edge of the flywheel.

    34. Satellite Orbiting. A satellite orbits the Earth in a circular path with a radius of 10,000 km at a constant speed of 7 km/s. Determine the angular velocity of the satellite in radians per hour.

    35. Speed of a Ferris Wheel. A Ferris wheel with a diameter of 100 meters takes 4 minutes to make one complete revolution. Calculate the linear velocity and angular velocity of a cabin at the edge of the Ferris wheel.

    36. Potter’s Wheel. A potter’s wheel moves from rest to an angular speed of 120 RPM in 5 seconds. Calculate the angular acceleration and the linear velocity at a point 15 cm from the center at the end of 5 seconds.

    37. Circular Saw. A circular saw blade has a radius of 15 cm and rotates at a speed of 3000 RPM. Find the angular velocity in radians per second and the linear velocity at the perimeter of the blade.

    Challenge Problems

    1. If the radius of a sector is doubled while the central angle remains the same, by what factor does the area increase?

    This page titled 4.4.2: Homework was last modified on Thu, 10 Jul 2025 21:53:54 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.