SECTION 6.1 Angles, Arc Length, and Circular Motion 395 121. Approximating the Circumference of Earth Eratosthenes of Cyrene (276–195 bc) was a Greek scholar who lived and worked in Cyrene and Alexandria. One day while visiting in Syene, he noticed that the Sun’s rays shone directly down a well. On this date 1 year later, in Alexandria, which is 500 miles due north of Syene, he measured the angle of the Sun to be about 7.2 degrees. See the figure. Use this information to approximate the radius and circumference of Earth. Syene 500 miles Alexandria 7.28 122. Designing a Little League Field For a 60-foot Little League Baseball field, the distance from home base to the nearest fence (or other obstruction) in fair territory should be a minimum of 200 feet. The commissioner of parks and recreation is making plans for a new 60-foot field. Because of limited ground availability, the minimum required distance to the outfield fence will be used.To increase safety, however, a 10-foot wide warning track will be installed on the inside of the fence. To further increase safety, the fence and warning track will extend both directions into foul territory. In total, the arc formed by the outfield fence (including the extensions into the foul territories) will be subtended by a central angle at home plate measuring 96 ,° as illustrated. (a) Determine the length of the outfield fence. (b) Determine the area of the warning track. Outfield Fence Home Base 200' 10' Warning Track 968 200' Foul Line 200' Foul Line H [Note: There is a 90° angle between the two foul lines. Then there are two 3° angles between the foul lines and the dashed lines shown. The angle between the two dashed lines outside the 200-foot foul lines is 96 .°] Source: www.littleleague.org 123. Challenge Problem Let the Dog Roam A dog is attached to a 9-foot rope fastened to the outside corner of a fencedin garden that measures 6 feet by 10 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander. Write the exact area in square feet.* 115. Computing the Speed of a River Current To approximate the speed of the current of a river, a circular paddle wheel with radius 4 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 10 revolutions per minute, what is the speed of the current? Express your answer in miles per hour. 116. Spin Balancing Tires A spin balancer rotates the wheel of a car at 480 revolutions per minute. If the diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour.At how many revolutions per minute should the balancer be set to test a road speed of 80 miles per hour? 117. The Cable Cars of San Francisco At the Cable Car Museum you can see the four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, driven by a rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in revolutions per minute. 118. Difference in Time of Sunrise Naples, Florida, is about 90 miles due west of Ft. Lauderdale. How much sooner would a person in Ft. Lauderdale first see the rising Sun than a person in Naples? [Hint: Consult the figure.When a person at Q sees the first rays of the Sun, a person at P is still in the dark.The person at P sees the first rays after Earth has rotated until P is at the location Q. Now use the fact that at the latitude of Ft. Lauderdale, in 24 hours an arc of length 2 3559 π( ) miles is subtended.] 3559 miles P Q Sun 90 miles Rotation of Earth Naples, P N S W E Fort Lauderdale, Q 119. Keeping Up with the Sun How fast would you have to travel on the surface of Earth at the equator to keep up with the Sun (that is, so that the Sun would appear to remain in the same position in the sky)? 120. Nautical Miles A nautical mile equals the length of the arc subtended by a central angle of 1 minute on a great circle† on the surface of Earth. See the figure. If the radius of Earth is taken as 3960 miles, express 1 nautical mile in terms of ordinary, or statute, miles. 4 ft †Any circle drawn on the surface of Earth that divides Earth into two equal hemispheres. South Pole North Pole Equator 1 minute 1 nautical mile *Courtesy of the Joliet Junior College Mathematics Department
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