Math Notes 4.1 Solution Key

HPC Notes 4.1 Arc Length, Area of a Sector, Linear/Angular Velocity S = rθ A = 1/2 r²θ Arc Length: part of the circumference Area of a Sector: part of the area of a circle Example 1: A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°. 240° = 4π/3 S = 4 * 4π/3 = 16π/3 Example 2: Find s. Same as Ex.1 Notice: s is the arc cut off by 240°, NOT 120°! Example 3: Find the central angle (in radians) that is formed by a radius of 12 ft and an arc length of 31 ft s = rθ 31 = 12θ θ = 31/12 Example 4: Find the distance between the cities. Assume that the Earth is a sphere of radius 4000 miles and the cities are on the same longitude (one city is due north of the other). r = 4000 θ = 26°8' + 31°46' = 57.9° → 57.9° * π/180° S = rθ = 4000 * 57.9° * π/180° ≈ 4042.18 miles Johannesburg, South Africa (26°8'S) Jerusalem, Israel (31°46'N) Example 5: Assuming the Earth is a sphere of radius 6378 km, what is the difference in latitudes of Lynchburg, Virginia and Myrtle Beach, South Carolina, where Lynchburg is 400 km due north of Myrtle Beach? Leave answer in DMS s = rθ 400 = 6378θ θ = 0.0627 rad 0.0627 * 180/π ≈ 3.6° Example 6: Find the area of the sector determined by the 100° central angle. A = 1/2 r²θ = 1/2 * 7² * 100° * π/180° ≈ 42.76 units² Write a formula relating distance, rate and time: d = rt r = d/t Linear Speed: How fast a particle is moving along a circular arc linear speed = s/t = rθ/t Angular Speed: How fast an angle changes angular speed = θ/t Equivalent ratios: 1 revolution = 2π radians = 360° = Circumference (2πr units or πd "units") Example 7: The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of the second hand in cm/s. r = 10.2 cm s = 2πr cm t = 60 sec s/t = (2π(10.2) cm) / 60 sec = 1.07 cm/sec Example 8: A 15-inch diameter tire on a car makes 9.3 revolutions per second. a. Find the angular speed of the tire in rad/sec   9.3 rev/1 sec * 2π rad/1 rev = 58.43 rad/sec b. Find the linear speed of the car in in/sec 2πr in/1 rev * 58.43 rev/1 sec = 438.23 in/sec Example 9: The circular blade on a saw has a diameter of 7.25 inches and rotates at 4800 revolutions per minute. a. Find the angular speed of the blade in rad/min   4800 rev/1 min * 2π rad/1 rev = 9600π rad/min ≈ 30159.3 rad/min b. Find the linear speed of the saw teeth (in ft/sec) as they contact the wood being cut. 30159.3 rad/min * 7.25π in/2π rad * 1 ft/12 in * 1 min/60 sec ≈ 151.84 ft/sec Example 10: A woman is riding a bicycle whose wheels are 30 inches in diameter. If the wheels rotate at 150 rpm, find the speed at which she is traveling in mi/hr. 150 rev/1 min * 30π in/1 rev * 60 min/1 hr * 1 ft/12 in * 1 mile/5280 ft ≈ 13.4 mph Example 11: A radial saw has a blade with a 6-inch radius. Suppose the blade spins at 1000 rpm. a. Find the angular speed of the blade in rad/min 1000 rev/1 min * 2π rad/1 rev = 2000π rad/min b. Find the linear speed of the saw teeth in ft/sec 2000π rad/min * 6π in/2π rad * 1 ft/12 in * 1 min/60 sec ≈ 157.08 ft/sec