1. { Applications of Radian Measure OBJECTIVE: Use angles to model and solve real-life problems

2.  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

3.  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

4. 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

5. 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

6.  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

7.  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

8.  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

9.  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

10.  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

11.  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