Concept Check Find the exact length of each arc

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Concept Check Find the exact length of each arc intercepted by the given central angle.

Explain how to find the degree measure of a central angle in a circle if both the radius and the length of the intercepted arc are known.

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 12.3 cm, u = 2p 3 radians

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 0.892 cm, u = 11p 10 radians

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 1.38 ft , u = 5p 6 radians

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 3.24 mi, u = 7p 6 radians

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 4.82 m, u = 60_

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 71.9 cm, u = 135_

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 15.1 in., u = 210_

Find the length to three significant digits of each arc intercepted by a central angle u in a circle of radius r. See Example 1. r = 12.4 ft, u = 330_

Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

Concept Check Radian measure simplifies many formulas, such as the formula for arc length, s = r u. Give the corresponding formula when u is measured in degrees instead of radians.

Panama City, Panama, 9 N, and Pittsburgh, Pennsylvania, 40 N

Farmersville, California, 36 N, and Penticton, British Columbia, 49 N

New York City, New York, 41 N, and Lima, Peru, 12 S

Halifax, Nova Scotia, 45 N, and Buenos Aires, Argentina, 34 S

Latitude of Madison Madison, South Dakota, and Dallas, Texas, are 1200 km apart and lie on the same north-south line. The latitude of Dallas is 33 N. What is the latitude of Madison?

Latitude of Toronto Charleston, South Carolina, and Toronto, Canada, are 1100 km apart and lie on the same north-south line. The latitude of Charleston is 33 N. What is the latitude of Toronto? Work each problem. See Examples 3 and 4.

Gear Movement Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through an angle of 300, through how many degrees does the larger gear rotate?

Gear Movement Repeat Exercise 27 for gear radii of 4.8 in. and 7.1 in. and for an angle of 315 for the smaller gear.

Rotating Wheels The rotation of the smaller wheel in the figure causes the larger wheel to rotate. Through how many degrees does the larger wheel rotate if the smaller one rotates through 60.0?

Rotating Wheels Repeat Exercise 29 for wheel radii of 6.84 in. and 12.46 in. and an angle of 150 for the smaller wheel.

Rotating Wheels Find the radius of the larger wheel in the figure if the smaller wheel rotates 80.0 when the larger wheel rotates 50.0.

Rotating Wheels Repeat Exercise 31 if the smaller wheel of radius 14.6 in. rotates 120 when the larger wheel rotates 60.

Pulley Raising a Weight Refer to the figure. (a) How many inches will the weight in the figure rise if the pulley is rotated through an angle of 71_ 50_? (b) Through what angle, to the nearest minute, must the pulley be rotated to raise the weight 6 in.?

Pulley Raising a Weight Find the radius of the pulley in the figure if a rotation of 51.6 raises the weight 11.4 cm.

Bicycle Chain Drive The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are rotated through 180? Assume the radius of the bicycle wheel is 13.6 in.

Car Speedometer The speedometer of Terrys Honda CR-V is designed to be accurate with tires of radius 14 in. (a) Find the number of rotations of a tire in 1 hr if the car is driven at 55 mph. (b) Suppose that oversize tires of radius 16 in. are placed on the car. If the car is now driven for 1 hr with the speedometer reading 55 mph, how far has the car gone? If the speed limit is 55 mph, does Terry deserve a speeding ticket? Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand.

Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand. 30 min

Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand. 40 min

Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand. 4.5 hr

Suppose the tip of the minute hand of a clock is 3 in. from the center of the clock. For each duration, determine the distance traveled by the tip of the minute hand. 6 1 2 Hr

Length of a Train A railroad track in the desert is 3.5 km away. A train on the track subtends (horizontally) an angle of 3_ 20_. Find the length of the train.

Distance to a Boat The mast of Brent Simons boat is 32 ft high. If it subtends an angle of 2_ 10_ , how far away is it? Concept Check Find the area of each sector.

Concept Check Find the area of each sector.

Concept Check Find the area of each sector.

Concept Check Find the area of each sector.

Concept Check Find the area of each sector.

Concept Check Find the measure (in degrees) of each central angle. The number inside the sector is the area.

Concept Check Find the measure (in degrees) of each central angle. The number inside the sector is the area.

Concept Check Find the measure (in radians) of each central angle. The number inside the sector is the area.

Concept Check Find the measure (in radians) of each central angle. The number inside the sector is the area.

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 29.2 m, u = 5p 6 radians

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 59.8 km, u = 2p 3 radians

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 30.0 ft, u = p 2 radians

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 90.0 yd, u = 5p 6 radians

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 12.7 cm, u = 81_

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 18.3 m, u = 125_

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 40.0 mi, u = 135_

Find the area of a sector of a circle having radius r and central angle u. Express answers to the nearest tenth. See Example 5. r = 90.0 km, u = 270_

Angle Measure Find the measure (in radians) of a central angle of a sector of area 16 in.2 in a circle of radius 3.0 in.

Radius Length Find the radius of a circle in which a central angle of p6 radian determines a sector of area 64 m2 .

Irrigation Area A center-pivot irrigation system provides water to a sector-shaped field as shown in the figure. Find the area of the field if u = 40.0_ and r = 152 yd.

Irrigation Area Suppose that in Exercise 61 the angle is halved and the radius length is doubled. How does the new area compare to the original area? Does this result hold in general for any values of u and r?

Arc Length A circular sector has an area of 50 in.2 . The radius of the circle is 5 in. What is the arc length of the sector?

Angle Measure In a circle, a sector has an area of 16 cm2 and an arc length of 6.0 cm. What is the measure of the central angle in degrees?

Measures of a Structure The figure illustrates Medicine Wheel, a Native American structure in northern Wyoming. There are 27 aboriginal spokes in the wheel, all equally spaced. (a) Find the measure of each central angle in degrees and in radians. (b) If the radius of the wheel is 76.0 ft, find the circumference. (c) Find the length of each arc intercepted by consecutive pairs of spokes. (d) Find the area of each sector formed by consecutive spokes.

Area Cleaned by a Windshield Wiper The Ford Model A, built from 1928 to 1931, had a single windshield wiper on the drivers side. The total arm and blade was 10 in. long and rotated back and forth through an angle of 95. The shaded region in the figure is the portion of the windshield cleaned by the 7-in. wiper blade. What is the area of the region cleaned?

Circular Railroad Curves In the United States, circular railroad curves are designated by the degree of curvature, the central angle subtended by a chord of 100 ft. Suppose a portion of track has curvature 42.0. (Source: Hay, W., Railroad Engineering, John Wiley and Sons.) (a) What is the radius of the curve? (b) What is the length of the arc determined by the 100-ft chord? (c) What is the area of the portion of the circle bounded by the arc and the 100-ft chord?

Land Required for a Solar-Power Plant A 300-megawatt solar-power plant requires approximately 950,000 m2 of land area to collect the required amount of energy from sunlight. If this land area is circular, what is its radius? If this land area is a 35 sector of a circle, what is its radius?

Area of a Lot A frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is that of finding the area when one or more of the boundary lines is the arc of a circle. Find the area (to two significant digits) of the lot shown in the figure. (Source: Anderson, J. and E. Michael, Introduction to Surveying, McGraw-Hill.)

Nautical Miles Nautical miles are used by ships and airplanes. They are different from statute miles, which equal 5280 ft. A nautical mile is defined to be the arc length along the equator intercepted by a central angle AOB of 1 min, as illustrated in the figure. If the equatorial radius of Earth is 3963 mi, use the arc length formula to approximate the number of statute miles in 1 nautical mile. Round your answer to two decimal places.

Circumference of Earth The first accurate estimate of the distance around Earth was done by the Greek astronomer Eratosthenes (276 195 b.c.), who noted that the noontime position of the sun at the summer solstice in the city of Syene differed by 7_ 12_ from its noontime position in the city of Alexandria. (See the figure.) The distance between these two cities is 496 mi. Use the arc length formula to estimate the radius of Earth. Then find the circumference of Earth. (Source: Zeilik, M., Introductory Astronomy and Astrophysics, Third Edition, Saunders College Publishers.)

Longitude Longitude is the angular distance (expressed in degrees) East or West of the prime meridian, which goes from the North Pole to the South Pole through Greenwich, England. Arcs of 1 longitude are 110 km apart at the equator, and therefore 15 arcs subtend 15(110) km, or 1650 km, at the equator. Because Earth rotates 15 per hr, longitude is found by taking the difference between time zones multiplied by 15. For example, if it is 12 noon where you are (in the United States) and 5 p.m. in Greenwich, you are located at longitude 5(15), or 75 W. (a) What is the longitude at Greenwich, England? (b) Use time zones to determine the longitude where you live.

Concept Check If the radius of a circle is doubled and the central angle of a sector is unchanged, how is the area of the sector changed?

Concept Check Give the formula for the area of a sector when the angle is measured in degrees.

Volume of a Solid Multiply the area of the base by the height to find a formula for the volume V of each solid.

Volume of a Solid Multiply the area of the base by the height to find a formula for the volume V of each solid.