## Algebra and Trigonometry

Mathematics

## Quiz 6 :

The Trigonometric Functions

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Q04 Q04 Q04

Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.

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Q14 Q14 Q14

The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R

_{ 0 }is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90^{ o }when the sun is in the east and = 0^{ o }when the sun is in the south. What percentage of R_{ 0 }is striking the wall when is equal to 30^{ o }and the sun is in the southeast?Free

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Q15 Q15 Q15

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 18 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (18, 0) and then rotates through an angle of 60

^{ o }, what is the new location of the hand?Free

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Q18 Q18 Q18

The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 19 days.

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Q19 Q19 Q19

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 32 occurs at 2 P.M., and the average temperature of 23 occurs 6 hours later.

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Q22 Q22 Q22

Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I

_{ o }is the maximum intensity. Approximate I in terms of I_{ o }for .Free

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Q23 Q23 Q23

A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

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Q29 Q29 Q29

Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.

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Q39 Q39 Q39

The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R

_{ 0 }is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90^{ o }when the sun is in the east and = 0^{ o }when the sun is in the south. What percentage of R_{ 0 }is striking the wall when is equal to 60^{ o }and the sun is in the southeast?Free

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Q40 Q40 Q40

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60

^{ o }, what is the new location of the hand?Free

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Multiple Choice

Q43 Q43 Q43

The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 29 days.

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Q44 Q44 Q44

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later.

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Q47 Q47 Q47

Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I

_{ o }is the maximum intensity. Approximate I in terms of I_{ o }for .Free

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Q48 Q48 Q48

A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

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Q54 Q54 Q54

Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.

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Multiple Choice

Q64 Q64 Q64

The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R

_{ 0 }is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90^{ o }when the sun is in the east and = 0^{ o }when the sun is in the south. What percentage of R_{ 0 }is striking the wall when is equal to 60^{ o }and the sun is in the southeast?Free

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Q65 Q65 Q65

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60

^{ o }, what is the new location of the hand?Free

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Q68 Q68 Q68

The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 23 days.

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Q69 Q69 Q69

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 34 occurs at 2 P.M., and the average temperature of 19 occurs 6 hours later.

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Q72 Q72 Q72

Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I

_{ o }is the maximum intensity. Approximate I in terms of I_{ o }for .Free

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Q73 Q73 Q73

A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

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Q78 Q78 Q78

The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon:

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Q84 Q84 Q84

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation while the relative humidity in percent could be expressed by where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

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Q88 Q88 Q88

The thickness of the ozone layer can be estimated using the formula where I

_{ 0 }is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, k is the absorption constant of ozone for that wavelength, and is the acute angle that the sunlight makes with the vertical. Suppose that for a wavelength of 3,055 10^{ - 8 }centimeter with k 1.93, I_{ o }/I is measured as 1.72 and . Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.Free

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Q89 Q89 Q89

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 16 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (16, 0) and then rotates through an angle of 60

^{ o }, what is the new location of the hand?Free

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Q91 Q91 Q91

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 17 , and the low temperature of -17 occurs at 4 A.M.

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Q99 Q99 Q99 A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

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Q103 Q103 Q103

The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon:

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Q109 Q109 Q109

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation while the relative humidity in percent could be expressed by where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

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Q113 Q113 Q113

The thickness of the ozone layer can be estimated using the formula where I

_{ 0 }is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, k is the absorption constant of ozone for that wavelength, and is the acute angle that the sunlight makes with the vertical. Suppose that for a wavelength of 3,055 10^{ - 8 }centimeter with k 1.86, I_{ o }/I is measured as 1.72 and . Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.Free

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Q114 Q114 Q114

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 18 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (18, 0) and then rotates through an angle of 60

^{ o }, what is the new location of the hand?Free

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Q116 Q116 Q116

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 25 , and the low temperature of -25 occurs at 4 A.M.

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Q124 Q124 Q124 A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

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Q128 Q128 Q128

The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon:

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Q134 Q134 Q134

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation while the relative humidity in percent could be expressed by where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

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Q138 Q138 Q138

The thickness of the ozone layer can be estimated using the formula where I

_{ 0 }is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, k is the absorption constant of ozone for that wavelength, and is the acute angle that the sunlight makes with the vertical. Suppose that for a wavelength of 3,055 10^{ - 8 }centimeter with k 1.93, I_{ o }/I is measured as 1.72 and . Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.Free

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Q139 Q139 Q139

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60

^{ o }, what is the new location of the hand?Free

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Q141 Q141 Q141

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 19 , and the low temperature of -19 occurs at 4 A.M.

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Q149 Q149 Q149 A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

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