{ Applications of Radian Measure OBJECTIVE: Use angles to model and solve real-life problems
Arc length: For a circle of radius, a central angle intercepts an arc of length given by, where is measured in radians. Note that if, then and the radian measure of equals the arc length Area of Sector: For a circle of radius, the area of a sector of the circle with central angle is, where is measured in radians Radian measure is used to simplify certain formulas
Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius. If is the length of the arc traveled in time, then the linear speed of the particle is Linear speed: Moreover, if is the angle (in radian measure) corresponding to the arc length, then the angular speed of the particle is Angular speed: Radian measure is used to simplify certain formulas
Finding Arc length 1. A circle has a radius 12.3 cm. Find the length of the arc intercepted by a central angle of 2. A circle has a radius of 4.82 m. Find the length of the arc intercepted by a central angle of
Finding Area of a Sector of a circle 3. Find the area of the sector with a radius of 9.0m and a central angle of 4. Find the area of the sector with a radius of 12.7cm and a central angle of
5. A sprinkler on a golf course fairway sprays over a distance of 70 feet and rotates through an angle of. Find the area of the fairway watered by the sprinkler. Area of a Sector of a Circle
6. Find the distance in kilometers between the pair of cities whose latitudes are given. Assume that the cities are on a north-south line and that the radius of the earth is 6400 km. Arc Length
7. The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face. Finding Angular and Linear Speeds
8. The blades of a wind turbine are 116 feet long. The propeller rotates at 15 revolutions per minute. A) Find the angular speed of the propeller in radians per minute B) Find the linear speed of the tips of the blades. Finding Angular and Linear Speeds
9. Suppose that point P is on a circle with a radius 10 centimeters and ray OP is rotating with angular speed of radians per second. A) Find the angle generated by P in 6 seconds B) Find the distance traveled by P along the circle in 6 seconds C) Find the linear speed of P Finding Angular and Linear Speeds
10. A satellite traveling in a circular orbit 1600 kilometers above the surface of the Earth takes two hours to make an orbit. Assume that the radius of the Earth is 6400 kilometers. A) Find the linear speed of the satellite B) Find the distance traveled in 4.5 hours Finding Angular and Linear Speeds