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Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Analytic Trigonometry with Applications, 11th Edition, Wiley, 2011.

EXERCISE 2.1

63. An angle that is inscribed in a circle of radius 3 m subtends an arc of length 2 m. Find its radian measure.

64. An angle that is inscribed in a circle of radius 4 m subtends an arc of length 5 m. Find its radian measure.

65. An angle that is inscribed in a circle of radius 6 ft subtends an arc of length 13 ft. Find its degree measure to the nearest degree.

66. An angle that is inscribed in a circle of radius 15 ft subtends an arc of length 34 ft. Find its degree measure to the nearest degree.

67. An angle of 3° is inscribed in a circle of radius 85 km. Find the length of the arc it subtends.

68. An angle of 27° is inscribed in a circle of radius 52 km. Find the length of the arc it subtends.

69. An angle of radian measure π/8 is inscribed in a circle and subtends an arc of length 42 cm. Find the radius of the circle.

70. An angle of radian measure 0.25 is inscribed in a circle and subtends an arc of length 26 cm. Find the radius of the circle.

71. What is the radian measure of the smaller angle made by the hands of a clock at 2:30 (see the figure)? Express the answer in terms of π and as a decimal fraction to two decimal places.

73. Pendulum. A clock has a pendulum 22 cm long. If it swings through an angle of 32°, how far does the bottom of the bob travel in one swing?

74. Pendulum. If the bob on the bottom of the 22 cm pendulum in Problem 73 traces a 9.5 cm arc on each swing, through  what  angle  (in  degrees)  does  the  pendulum rotate on each swing?

75. Engineering. Oil is pumped from some wells using a donkey pump as shown in the figure. Through how many degrees must an arm with a 72 in. radius rotate to produce a 24 in. vertical stroke at the pump down in the ground? Note that a point at the end of the arm must travel through a 24 in. arc to produce a 24 in. vertical stroke at the pump.

76. Engineering. In Problem 75, find the arm length r that would produce an 18 in. vertical stroke while rotating through 21°.

77. Bioengineering. A particular woman, when standing and facing forward, can swing an arm in a plane perpendicular to her shoulders through an angle of 3 p> 2 rad (see the figure). Find the length of the arc (to the nearest centimeter) her fingertips trace out for one complete swing of her arm. The length of her arm from the pivot point in her shoulder to the end of her longest finger is 54.3 cm while it is kept straight and her fingers are extended with palm facing in.

78. Bioengineering. A particular man, when standing and facing forward, can swing a leg through an angle of 2π/3 rad (see the figure). Find the length of the arc (to the nearest centimeter) his heel traces out for one complete swing of the leg. The length of his leg from the pivot point in his hip to the bottom of his heel is 102 cm while his leg is kept straight and his foot is kept at right angles to his leg.

79. Astronomy. The sun is about from the earth. If the angle subtended by the diameter of the sun on the surface of the earth is approximately what is the diameter of the sun?

80. Surveying. If a natural gas tank 5.000 km away subtends an angle of 2.44°, approximate its height to the nearest meter (see Problem 79).

81. Photography. The angle of view for a 300 mm telephoto lens is 8° (see the figure). At 1,250 ft, what is the approximate width of the field of view? Use an arc length to approximate the chord length to the nearest foot.

82. Photography. The angle of view for a 1,000 mm telephoto lens is 2.5° (see the figure). At 865 ft, what is the approximate width of the field of view? Use an arc length to approximate the chord length to the nearest foot.

83. Spy Satellites. Some spy satellites have cameras that can distinguish objects that subtend angles of as little as 5 × 10-7 rad. If such a satellite passes over a particular country at an altitude of 250 mi, how small an object can the camera distinguish? Give the answer in meters to one decimal place and also in inches to one decimal place.

85. Astronomy. Assume that the earth’s orbit is circular. A line from the earth to the sun sweeps out an angle of how many radians in 1 week? Express the answer in terms of p and as a decimal fraction to two decimal places. (Assume exactly 52 weeks in a year.)

87. Astronomy. In measuring time, an error of 1 sec per day may not seem like a lot. But suppose a clock is in error by at most 1 sec per day. Then in 1 year the accumulated error could be as much as 365 sec. If we assume the earth’s orbit about the sun is circular, with a radius of 9.3 × 107 mi, what would be the maximum error (in miles) in computing the distance the earth travels in its orbit in 1 year?

88. Astronomy. Using the clock described in Problem 87, what would be the maximum error (in miles) in computing  the  distance  that  Venus  travels  in  a  “Venus  year”? Assume Venus’s orbit around the sun is circular, with a radius of 6.7 × 107 mi, and  that  Venus  completes  one orbit (a “Venus year”) in 224 earth days.

89. Geometry. A sector of a circle has an area of 52.39 ft2 and a radius of 10.5 ft. Calculate the perimeter of the sector to the nearest foot.

90. Geometry. A sector of a circle has an area of 145.7 cm2 and a radius of 8.4 cm. Calculate the perimeter of the sector to the nearest centimeter.

93. Engineering. Rotation of a drive wheel causes a shaft to rotate (see the figure). If the drive wheel completes 3 revolutions, how many revolutions will the shaft complete? Through how many radians will the shaft turn? Compute answers to one decimal place.

94. Engineering. In Problem 93, find the radius (to the nearest millimeter) of the drive wheel required for the 12 mm shaft to make 7 revolutions when the drive wheel makes 3 revolutions.

95. Radians and Arc Length. A bicycle wheel of diameter 32 in. travels a distance of 20 ft. Find the angle (to the nearest degree) swept out by one of the spokes.

96. Radians and Arc Length. A bicycle has a front wheel with a diameter of 24 cm and a back wheel with a diameter of 60 cm. Through what angle (in radians) does the front wheel turn if the back wheel turns through 12 rad?

97. Cycling. A bicycle has 28 in. diameter tires. The largest front gear (at the pedals) has 48 teeth and the smallest has 20 teeth. The largest rear gear has 32 teeth and the smallest has 11 teeth. Determine the maximum distance traveled (to the nearest inch) in one revolution of the pedals.

98. Cycling. Refer to Problem 97. Determine the minimum distance traveled (to the nearest inch) in 1 revolution of the pedals.


EXERCISE 2.2


19. Engineering. A 16 mm diameter shaft rotates at 1,500 rps (revolutions per second). Find the speed of a point on its surface (to the nearest meter per second).

20. Engineering. A 6 cm diameter shaft rotates at 500 rps. Find the speed of a point on its surface (to the nearest meter per second).

21. Space Science. An earth satellite travels in a circular orbit at 20,000 mph. If the radius of the orbit is 4,300 mi, what angular velocity (in radians per hour, to three significant digits) is generated?

22. Engineering. A bicycle is ridden at a speed of 7.0 m/sec. If the wheel diameter is 64 cm, what is the angular velocity of the wheel in radians per second?

23. Physics. The velocity of sound in air is approximately 335.3 m/sec. If an airplane has a 3.000 m diameter propeller, at what angular velocity will its tip pass through the sound barrier?

24. Physics. If an electron in an atom travels around the nucleus in a circular orbit (see the figure on the next page) at 8.11 × 106 cm/sec, what angular velocity (in radians per second) does it generate, assuming the radius of the orbit is 5.00 × 10-9 cm?

25. Astronomy. The earth revolves about the sun in an orbit that is approximately circular with a radius of 9.3 × 107 mi (see the figure). The radius of the orbit sweeps out an angle with what exact angular velocity (in radians per hour)? How fast (to the nearest hundred miles per hour) is the earth traveling along its orbit?

26. Astronomy. Take into consideration only the daily rotation of the earth to find out how fast (in miles per hour) a person halfway between the equator and North Pole would be moving. The radius of the earth is approx. 3,964 mi, and a daily rotation takes 23.93 hr.

27. Astronomy. Jupiter makes one full revolution about its axis every 9 hr 55 min. If Jupiter’s equatorial diameter is 88,700 mi: (A) What is its angular velocity relative to its axis of rotation (in radians per hour)? (B) What is the linear velocity of a point on Jupiter’s equator?

28. Astronomy. The sun makes one full revolution about its axis every 27.0 days. Assume 1 day = 24 hr. If its equatorial diameter is 865,400 mi: (A) What is the sun’s angular velocity relative to its axis of rotation (in radians per hour)? (B) What  is  the  linear  velocity  of  a  point  on  the  sun’s equator?

29. Space Science. For an earth satellite to stay in orbit over a given stationary spot on earth, it must be placed in orbit 22,300 mi above the earth’s surface (see the figure). It will then take the satellite the same time to complete one orbit as the earth, 23.93 hr. Such satellites are called geostationary satellites and are used for communications and tracking space shuttles. If the radius of the earth is 3,964 mi, what is the linear velocity of a geostationary satellite?

30. Astronomy. From 1979 to 1999 the planet Neptune was farther from the sun than Pluto. (After 1999, for 228 years Pluto will be farther away.) If Neptune is 2.795 ×109 mi from the sun, and completes one orbit in 164 years, find its linear velocity in miles per hour.

31. Space Science The earth rotates on its axis once every 23.93 hr, and a space shuttle revolves round the earth in the plane of the earth’s equator once every 1.51 hr. Both are rotating in the same direction (see the figure on the next page). What is the length of time between consecutive passages of the shuttle over a particular point P on the equator?

32. Astronomy One of the moons of Jupiter rotates around the planet in its equatorial plane once every 42 hr 30 min. Jupiter rotates around its axis once every 9 hr 55 min. Both are rotating in the same direction. What is the length of time between consecutive passages of the moon over the same point on Jupiter’s equator?

33. Precalculus: Rotating Beacon. A beacon light 15 ft from a wall rotates clockwise at the rate of exactly 1 rps (see the figure). To answer the following questions, start counting time (in seconds) when the light spot is at C. (A) Describe how you can represent θ in terms of time t; then represent θ in terms of t. (B) Describe how you can represent a, the distance the light travels along the wall, in terms of t; then represent a in terms of t.

34. Precalculus: Rotating Beacon. (A) Referring  to  the  rotating  beacon  in  Problem  33, describe  how  you  can  represent  c, the  length  of  the light beam, in terms of t; then represent c in terms of t. (B) Complete Table 2, relating c and t, to two decimal places.  (If your calculator has a table-producing capability, use it.) What does Table 2 seem to tell you about the rate of change of the length of the light beam as t increases from 0.00 to 0.24? What happens to c when t = 0.25?

35. Cycling. A racing bicycle has 28 in. diameter tires. The front gears (at the pedals) have 53 teeth and 39 teeth. The largest rear gear has 21 teeth and the smallest has 11 teeth. Determine the maximum speed of the bicycle (to the nearest tenth of a mile per hour) if the cyclist’s cadence is 90 revolutions of the pedals per minute.

36. Cycling. Refer to Problem 35. When the bicycle is in its highest gear, what cadence (to the nearest rpm) is required to maintain a speed of 26.5 miles per hour? (“Highest gear” refers to the largest ratio of teeth on the front gear to teeth on the rear gear.)

 

EXERCISE 2.3

81. Solar Energy Light intensity I on a solar cell changes with the angle of the sun and is given by the formula in the following figure. Find the intensity in terms of the constant k for θ = 0°, θ = 20°, θ = 40°, θ = 60°, and θ = 80°. Compute each answer to two decimal places.

82. Solar Energy. In Problem 81, at what angle will the light intensity I be 50% of the vertical intensity?

83. The amount of heat energy E from the sun received per square meter per unit of time in a given region on the surface of the earth is approximately proportional to the cosine of the angle θ that the sun makes with the vertical (see the figure). That is, E = k cos θ, where k is the constant of proportionality for a given region. For a region with a latitude of 40° N, compare the energy received at the summer solstice (θ = 15°) with the energy received at the winter solstice (θ = 63°). Express answers in terms of k to two significant digits.

84. For a region with a latitude of 32° N, compare the energy received at the summer solstice (θ = 8°) with the energy received at the winter solstice (θ = 55°). Refer to Problem 83, and express answers in terms of k to two significant digits.

85. Precalculus: Calculator Experiment. It can be shown that the area of a polygon of n equal sides inscribed in a circle of radius 1 is given by An = n/2 sin(360/n). (A) Complete the table, giving An to five decimal places (B) As n gets larger and larger, what number does seem to approach? (C) Will an inscribed polygon ever be a circle for any n, however large? Explain.

86. Precalculus: Calculator Experiment. It can be shown that the area of a polygon of n equal sides circumscribed around a circle of radius 1 is given by An = n tan(180/n). (A) Complete the table, giving to five decimal places: (B) As n gets larger and larger, what number does seem to approach? (C) Will a circumscribed polygon ever be a circle for any n, however large? Explain.

87. Engineering. The figure shows a piston connected to a wheel that turns at 10 revolutions per second (rps). If P is at (1, 0) when t = 0, then θ = 20πt, where t is time in seconds. Show that x = a + √(52 – b2) = cos 20πt√(25 – (sin20πt)2)

88. Engineering. In Problem 87, find the position (to two decimal places) of the piston (the value of x) for t = 0 and t = 0.01 sec.

89. Alternating Current. An alternating current generator produces an electric current (measured in amperes) that is described by the equation I = 35 sin(48πt  - 12π) where t is time in seconds. (See Example 6 and the figure that follows.) What is the current I when t = 0.13 sec?

90. Alternating Current. What is the current I in Problem 89 when t = 0.310 sec?

91. Precalculus: Angle of Inclination. The slope of a non-vertical line passing through points P1 = (x1, y1) and P2 = (x2, y2) is given by the formula
Slope = m = (y2 – y1)/(x2 – x1)
The angle θ that the line L makes with the x axis, is called the angle of inclination of the line L (see the figure). Thus, Slope = m = tan θ, 0° ≤ θ ≤ 180°. (A) Compute the slopes (to two decimal places) of the lines with angles of inclination 63.5° and 172°. (B) Find the equation of a line passing through with an angle of inclination 143°. Write the answer in the form with m and b to two decimal places.

92. Precalculus: Angle of Inclination. Refer to Problem 91. (A) Compute the slopes (to two decimal places) of the lines with angles of inclination 89.2° and 179°. (B) Find the equation of a line passing through (7, -4) with an angle of inclination 101°. Write the answer in the form y = mx + b, with m and b to two decimal places.

 

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