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All Textbook Solutions for Precalculus: Mathematics for Calculus - 6th Edition
50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68ETravel Distance A cars wheels are 28 in. in diameter. How far (in mi.) will the car travel if its wheels revolve 10,000 times without slipping?Wheel Revolutions How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?Latitudes Pittsburgh, Pennsylvania, and Miami, Florida, lie approximately on the same meridian. Pittsburgh has a latitude of 40.5N, and Miami has a latitude of 25.5N. Find the distance between these two cities. (The radius of the earth is 3960 mi.)72E73ECircumference of the Earth The Greek mathematician Eratosthenes (ca. 276195 b.c.) measured the circumference of the earth from the following observations. He noticed that on a certain day the sun shone directly down a deep well in Syene (modern Aswan). At the same time in Alexandria, 500 miles north (on the same meridian), the rays of the sun shone at an angle of 7.2 to the zenith. Use this information and the figure to find the radius and circumference of the earth.75EIrrigation An irrigation system uses a straight sprinkler pipe 300 ft long that pivots around a central point as shown. Because of an obstacle the pipe is allowed to pivot through 280 only. Find the area irrigated by this system.Windshield Wipers The top and bottom ends of a windshield wiper blade are 34 in. and 14 in., respectively, from the pivot point. While in operation, the wiper sweeps through 135. Find the area swept by the blade.The Tethered Cow A cow is tethered by a 100-ft rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze.Fan A ceiling fan with 16-in. blades rotates at 45 rpm. (a) Find the angular speed of the fan in rad/min. (b) Find the linear speed of the tips of the blades in in./min.Radial Saw A radial saw has a blade with a 6-in. radius. Suppose that the blade spins at 1000 rpm. (a) Find the angular speed of the blade in rad/min. (b) Find the linear speed of the sawteeth in ft/s.Winch A winch of radius 2 ft is used to lift heavy loads. If the winch makes 8 revolutions every 15 s, find the speed at which the load is rising.Speed of a Car The wheels of a car have radius 11 in. and are rotating at 600 rpm. Find the speed of the car in mi/h.Speed at the Equator The earth rotates about its axis once every 23 h 56 min 4 s, and the radius of the earth is 3960 mi. Find the linear speed of a point on the equator in mi/h.Truck Wheels A truck with 48-in.-diameter wheels is traveling at 50 mi/h. (a) Find the angular speed of the wheels in rad/min. (b) How many revolutions per minute do the wheels make?Speed of a Current To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius 0.20 m and rotates at 100 rpm, find the speed of the current in m/s.Bicycle Wheel The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 40 rpm. (a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)Conical Cup A conical cup is made from a circular piece of paper with radius 6 cm by cutting out a sector and joining the edges as shown below. Suppose = 5/3. (a) Find the circumference C of the opening of the cup. (b) Find the radius r of the opening of the cup. [Hint: Use C = 2r.] (c) Find the height h of the cup. [Hint: Use the Pythagorean Theorem.] (d) Find the volume of the cup.Conical Cup In this exercise we find the volume of the conical cup in Exercise 93 for any angle . (a) Follow the steps in Exercise 93 to show that the volume of the cup as a function of is V()=922422,02 (b) Graph the function V. (c) For what angle is the volume of the cup a maximum?89E90EA right triangle with an angle is shown in the figure. (a) Label the opposite and adjacent sides of and the hypotenuse of the triangle. (b) The trigonometric functions of the angle are defined as follows: sin=cos=tan= (c) The trigonometric ratios do not depend on the size of the triangle. This is because all right triangles with the same acute angle are __________.The reciprocal identities state that csc=1sec=1cot=13E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46EHeight of a Building The angle of elevation to the top of the Empire State Building in New York is found to be 11 from the ground at a distance of 1 mi from the base of the building. Using this information, find the height of the Empire State Building.Gateway Arch A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 35,000 ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is 22. (a) What is the distance between the plane and the arch? (b) What is the distance between a point on the ground directly below the plane and the arch?Deviation of a Laser Beam A laser beam is to be directed toward the center of the moon, but the beam strays 0.5 from its intended path. (a) How far has the beam diverged from its assigned target when it reaches the moon? (The distance from the earth to the moon is 240,000 mi.) (b) The radius of the moon is about 1000 mi. Will the beam strike the moon?Distance at Sea From the top of a 200-ft lighthouse, the angle of depression to a ship in the ocean is 23. How far is the ship from the base of the lighthouse?Leaning Ladder A 20-ft ladder leans against a building so that the angle between the ground and the ladder is 72. How high does the ladder reach on the building?52EElevation of a Kite A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level and estimates the angle of elevation of the kite to be 50. If the string is 450 ft long, how high is the kite above the ground?Determining a Distance A woman standing on a hill sees a flagpole that she knows is 60 ft tall. The angle of depression to the bottom of the pole is 14, and the angle of elevation to the top of the pole is 18. Find her distance x from the pole.Height of a Tower A water tower is located 325 ft from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is 39 and that the angle of depression to the bottom of the tower is 25. How tall is the tower? How high is the window?56EDetermining a Distance If both cars in Exercise 62 are on one side of the plane and if the angle of depression to one car is 38 and that to the other car is 52, how far apart are the cars?Height of a Balloon A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 20 and 22. How high is the balloon?Height of a Mountain To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be 32. One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is 35. Estimate the height of the mountain.Height of Cloud Cover To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75 from the horizontal. An observer 600 m away measures the angle of elevation to the spot of light to be 45. Find the height h of the cloud cover.Distance to the Sun When the moon is exactly half full, the earth, moon, and sun form a right angle (see the figure). At that time the angle formed by the sun, earth, and moon is measured to be 89.85. If the distance from the earth to the moon is 240,000 mi, estimate the distance from the earth to the sun.Distance to the Moon To find the distance to the sun as in Exercise 67, we needed to know the distance to the moon. Here is a way to estimate that distance: When the moon is seen at its zenith at a point A on the earth, it is observed to be at the horizon from point B (see the following figure). Points A and B are 6155 mi apart, and the radius of the earth is 3960 mi. (a) Find the angle in degrees. (b) Estimate the distance from point A to the moon.63EParallax To find the distance to nearby stars, the method of parallax is used. The idea is to find a triangle with the star at one vertex and with a base as large as possible. To do this, the star is observed at two different times exactly 6 months apart, and its apparent change in position is recorded. From these two observations E1SE2 can be calculated. (The times are chosen so that E1SE2 is as large as possible, which guarantees that E1OS is 90.) The angle E1SO is called the parallax of the star. Alpha Centauri, the star nearest the earth, has a parallax of 0.000211. Estimate the distance to this star. (Take the distance from the earth to the sun to be 9.3 107 mi.)65E66EIf the angle is in standard position and P(x, y) is a point on the terminal side of , and r is the distance from the origin to P, then sin=cos=tan=The sign of a trigonometric function of depends on the _____ in which the terminal side of the angle lies. In Quadrant II, sin is _______ (positive / negative). In Quadrant III, cos is _______ (positive / negative). In Quadrant IV, sin is _______(positive / negative).3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62EHeight of a Rocket A rocket fired straight up is tracked by an observer on the ground 1 mi away. (a) Show that when the angle of elevation is , the height of the rocket (in ft) is h = 5280 tan . (b) Complete the table to find the height of the rocket at the given angles of elevation.Rain Gutter A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle . (See the figure on the next page.) (a) Show that the cross-sectional area of the gutter is modeled by the function A()=100sin+100sincos (b) Graph the function A for 0 /2. (c) For what angle is the largest cross-sectional area achieved?Wooden Beam A rectangular beam is to be cut from a cylindrical log of diameter 20 cm. The figures show different ways this can be done. (a) Express the cross-sectional area of the beam as a function of the angle in the figures. (b) Graph the function you found in part (a). (c) Find the dimensions of the beam with largest cross-sectional area.66EThrowing a Shot Put The range R and height H of a shot put thrown with an initial velocity of v0 ft/s at an angle are given by R=v02sin(2)gH=v02sin22g On the earth g = 32 ft/s2, and on the moon g = 5.2 ft/s2. Find the range and height of a shot put thrown under the given conditions. (a) On the earth with v0 = 12 ft/s and = /6 (b) On the moon with v0 = 12 ft/s and = /6Sledding The time in seconds that it takes for a sled to slide down a hillside inclined at an angle is t=d16sin where d is the length of the slope in feet. Find the time it takes to slide down a 2000-ft slope inclined at 30.Beehives In a beehive each cell is a regular hexagonal prism, as shown in the figure. The amount of wax W in the cell depends on the apex angle and is given by W=3.020.38cot+0.65csc Bees instinctively choose so as to use the least amount of wax possible. (a) Use a graphing device to graph W as a function of for 0 . (b) For what value of does W have its minimum value? [Note: Biologists have discovered that bees rarely deviate from this value by more than a degree or two.]Turning a Comer A steel pipe is being carried down a hallway that is 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. (a) Show that the length of the pipe in the figure is modeled by the function L()=9csc+6sec (b) Graph the function L for 0 /2. (c) Find the minimum value of the function L. (d) Explain why the value of L you found in part (c) is the length of the longest pipe that can be carried around the corner.Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation of a rainbow is always the same. It can be shown that = 4 2. Where sin=ksin and = 59.4 and k = 1.33 is the index of refraction of water. Use the given information to find the angle of elevation of a rainbow. [Hint: Find sin , then use the SIN1 key on your calculator to find .] (For a mathematical explanation of rainbows see Calculus Early Transcendentals, 7th Edition, by James Stewart, page 282.)72E73E1EIn the triangle shown we can find the angle as follows. (a) =sin1 (b) =cos1 (c) =tan13E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36ELeaning Ladder A 20-ft ladder is leaning against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle of elevation of the ladder? How high does the ladder reach on the building?38EHeight of the Space Shuttle An observer views the space shuttle from a distance of 2 mi from the launch pad. (a) Express the height of the space shuttle as a function of the angle of elevation . (b) Express the angle of elevation as a function of the height h of the space shuttle.Height of a Pole A 50-ft pole casts a shadow as shown in the figure. (a) Express the angle of elevation of the sun as a function of the length s of the shadow. (b) Find the angle of elevation of the sun when the shadow is 20 ft long.Height of a Balloon A 680-ft rope anchors a hot-air balloon as shown in the figure. (a) Express the angle as a function of the height h of the balloon. (b) Find the angle if the balloon is 500 ft high.View from a Satellite The figures on the next page indicate that the higher the orbit of a satellite, the more of the earth the satellite can see. Let , s, and h be as in the figure, and assume that the earth is a sphere of radius 3960 mi. (a) Express the angle as a function of h. (b) Express the distance s as a function of . (c) Express the distance s as a function of h. [Hint: Find the composition of the functions in parts (a) and (b).] (d) If the satellite is 100 mi above the earth, what is the distance s that it can see? (e) How high does the satellite have to be to see both Los Angeles and New York, 2450 mi apart?Surfing the Perfect Wave For a wave to be surfable, it cant break all at once. Robert Gura and Tony Bowen have shown that a wave has a surfable shoulder if it hits the shoreline at an angle given by =sin1(12n+1tan) where is the angle at which the beach slopes down and where n = 0, 1, 2, . (a) For = 10, find when n = 3. (b) For = 15, find when n = 2, 3, and 4. Explain why the formula does not give a value for when n = 0 or 1.44EIn triangle ABC with sides a, b, and c the Law of Sines states that ==2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32ETracking a Satellite The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 50 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0 and 84.2, respectively. (a) How far is the satellite from station A? (b) How high is the satellite above the ground?Flight of a Plane A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 5 mi apart, to be 32 and 48, as shown in the figure. (a) Find the distance of the plane from point A. (b) Find the elevation of the plane.35EDistance Across a Lake Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such that CAB = 48.6. He also measures CA as 312 ft and CB as 527 ft. Find the distance between A and B.The Leaning Tower of Pisa The bell lower of the cathedral in Pisa, Italy, leans 5.6 from the vertical. A tourist stands 105 m from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be 29.2. Find the length of the tower to the nearest meter.Radio Antenna A short-wave radio antenna is supported by two guy wires, 165 ft and 180 ft long. Each wire is attached to the top of the antenna and anchored to the ground at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of 67 with the ground. How far apart are the anchor points?Height of a Tree A tree on a hillside casts a shadow 215 ft down the hill. If the angle of inclination of the hillside is 22 to the horizontal and the angle of elevation of the sun is 52, find the height of the tree.Length of a Guy Wire A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58. A guy wire is to be attached to the top of the tower and to the ground, 100 m downhill from the base of the tower. The angle in the figure is determined to be 12. Find the length of cable required for the guy wire.Calculating a Distance Observers at P and Q are located on the side of a hill that is inclined 32 to the horizontal, as shown. The observer at P determines the angle of elevation to a hot-air balloon to be 62. At the same instant the observer at Q measures the angle of elevation to the balloon to be 71. If P is 60 m down the hill from Q, find the distance from Q to the balloon.Calculating an Angle A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill it is observed that the angle formed between the top and base of the tower is 8. Find the angle of inclination of the hill.Distances to Venus The elongation of a planet is the angle formed by the planet, earth, and sun (see the figure). It is known that the distance from the sun to Venus is 0.723 AU (see Exercise 71 in Section 6.2). At a certain time the elongation of Venus is found to be 39.4. Find the possible distances from the earth to Venus at that time in astronomical units (AU).Soap Bubbles When two bubbles cling together in midair, their common surface is part of a sphere whose center D lies on the line passing through the centers of the bubbles (see the figure). Also, ACB and ACD each have measure 60. (a) Show that the radius r of the common face is given by r=abab [Hint: Use the Law of Sines together with the fact that an angle and its supplement 180 have the same sine.] (b) Find the radius of the common face if the radii of the bubbles are 4 cm and 3 cm. (c) What shape does the common face take if the two bubbles have equal radii?45EFor triangle ABC with sides a, b, and c the Law of Cosines states c2=In which of the following cases must the Law of Cosines be used to solve a triangle? ASASSSSASSSA3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E