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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

1. The domain of a logistic growth function cannot be the set of real numbers. 2. A logistic growth function will always have an x-intercept. 3. The graph of f(x) = 4/1 + 6e-2x + 5 is the graph of g(x) = 4/1 + 6e-2x shifted to the right five units. Determine whether the statement is true or false.
The graph of a Gaussian model will never have an x-intercept. True or false.
Identify each model as exponential growth, exponential decay, Gaussian, linear, logarithmic, logistic growth, quadratic, or none of the above. Explain your reasoning.To work an extended application analyzing the sales per share for Kohl's Corporation from 1999 through 2014, visit this text's
Find the missing values assuming continuously compounded interest.1.2.
Evaluate the function at the given value of x. Round your result to three decimal places. 1. f (x) = 0.3x, x = 1.5 2. f (x) = 30x, x = √3 3. f (x) = 2x, x = 2/3 4. f (x) = (1/2)2x, x = π 5. f (x) = 7(0.2x), x = −√11 6. f (x) = −14(5x), x = −0.8
Match the function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b) (c) (d) (e) (f) 1. y = 3eˆ’2x/3 2. y = 4e2x/3 3. y = ln(x + 3) 4. y = 7 ˆ’ log(x + 3)
Find the exponential model y = aebx that fits the points (0, 2) and (4, 3).
A species of bat is in danger of becoming extinct. Five years ago, the total population of the species was 2000. Two years ago, the total population of the species was 1400. What was the total population of the species one year ago?
The test scores for a biology test follow the normal distribution y = 0.0499e-(x - 7)2/128, 40 ≤ x ≤ 100 Where x is the test score. Use a graphing utility to graph the equation and estimate the average test score.
In a typing class, the average number N of words per minute typed after t weeks of lessons is N = 157 / (1 + 5.4e−0.12t). Find the time necessary to type (a) 50 words per minute and (b) 75 words per minute.
The relationship between the number of decibels β and the intensity of a sound I (in watts per square meter) is β = 10 log(I/10−12). Find the intensity I for each decibel level β (a) β = 60 (b) β = 135 (c) β = 1
Consider the graph of y = ekt. Describe the characteristics of the graph when k is positive and when k is negative.
Use a One-to-One Property to solve the equation for x. 1. (1/3)x−3 = 9 2. 3x+3 = 1/81 3. e3x−5 = e7 4. e8−2x = e−3
Describe the transformation of the graph of f that yields the graph of g. 1. f (x) = 5x, g(x) = 5x + 1 18. f (x) = 6x, g(x) = 6x+1 19. f (x) = 3x, g(x) = 1 − 3x 20. f (x) = (1/2)x, g(x) = −(1/2)x+2
Evaluate f (x) = ex at the given value of x. Round your result to three decimal places. 1. x = 3.4 2. x = −2.5 3. x = 3/5 4. x = 2/7
Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 1. h(x) = e−x/2 2. h(x) = 2 − e−x/2 3. f (x) = ex+2 4. s(t) = 4et−1
After t years, the value V of a car that originally cost $23,970 is given by V (t) = 23,970 (3/4)t. (a) Use a graphing utility to graph the function. (b) Find the value of the car 2 years after it was purchased. (c) According to the model, when does the car depreciate most rapidly? Is this
Complete the table by finding the balance A when P dollars is invested at rate r for t years and compounded n times per year.1. P = $5000, r = 3%, t = 10 years 2. P = $4500, r = 2.5%, t = 30 years
Evaluate the logarithm at the given value of x without using a calculator. 1. f (x) = log x, x = 1000 2. g(x) = log9 x, x = 3 3. g(x) = log2 x, x = 14 4. f (x) = log3 x, x = 1/81
Use a One-to-One Property to solve the equation for x. 1. log4 (x + 7) = log4 14 2. log8 (3x − 10) = log8 5 3. ln (x + 9) = ln 4 4. log (3x − 2) = log 7
Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph. 1. g(x) = log7 x 2. f (x) = log x/3
Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph. 1. f (x) = ln x + 6 2. f (x) = ln x - 5
The formula M = m − 5 log (d/10) gives the distance d (in parsecs) from Earth to a star with apparent magnitude m and absolute magnitude M. The star Rasalhague has an apparent magnitude of 2.08 and an absolute magnitude of 1.3. Find the distance from Earth to Rasalhague.
Evaluate the logarithm using the change-of-base formula (a) With common logarithms and (b) With natural logarithms. Round your results to three decimal places. 1. log2 6 2. log12 200 3. log1/2 5 4. log4 0.75
Use the properties of logarithms to write the logarithm in terms of log2 3 and log2 5. 1. log2 5/3 2. log2 45 3. log2 9/5 4. log2 20/9
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 1. log 7x2 2. log 11x3 3. log3 9/√x 4. log7 3√ x / 19
Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 1. f (x) = 4−x + 4 2. f (x) = 2.65x−1 3. f (x) = 5x−2 + 4 4. f (x) = 2x−6 − 5 5. f (x) = (1/2)−x + 3 6. f (x) = (1/8)x+2 - 5
Condense the expression to the logarithm of a single quantity. 1. ln 7 + ln x 2. log2 y − log2 3 3. log x - ½ log y 4. 3 ln x + 2 ln(x + 1) 5. ½ log3 x − 2 log3(y + 8) 6. 5 ln(x − 2) − ln(x + 2) − 3 ln x
The time t (in minutes) for a small plane to climb to an altitude of h feet is modeled by t = 50 log [18,000/(18,000 − h)] where 18,000 feet is the plane's absolute ceiling. (a) Determine the domain of the function in the context of the problem. (b) Use a graphing utility to graph the function
Solve for x. 1. 5x = 125 2. 6x = 1/216 3. ex = 3 4. log x − log 5 = 0 5. ln x = 4 6. ln x = −1.6
Solve the exponential equation algebraically. Approximate the result to three decimal places. 1. e4x = ex2+3 2. e3x = 25 3. 2x − 3 = 29 4. e2x − 6ex + 8 = 0
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. 1. ln 3x = 8.2 2. 4 ln 3x = 15
A 7 1/4 -inch circular power saw blade rotates at 5200 revolutions per minute. (a) Find the angular speed of the saw blade in radians per minute. (b) Find the linear speed (in feet per minute) of the saw teeth as they contact the wood being cut.
A carousel with a 50-foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed (in feet per minute) of the platform rim of the carousel.
A Blu-ray disc is approximately 12 centimeters in diameter. The drive motor of a Blu-ray player is able to rotate up to 10,000revolutions per minute. (a) Find the maximum angular speed (in radians per second) of a Blu-ray disc as it rotates. (b) Find the maximum linear speed (in meters per
A computerized spin balance machine rotates a 25-inch-diameter tire at 480 revolutions per minute. (a) Find the road speed (in miles per hour) at which the tire is being balanced. (b) At what rate should the spin balance machine be set so that the tire is being tested for 55 miles per hour?
The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4inches, 2inches, and 14inches, respectively. A cyclist pedals at a rate of 1revolution per second.(a) Find the speed of the bicycle in feet per second and miles per hour. (b) Use your result from
1. A car's rear windshield wiper rotates 125°. The total length of the wiper mechanism is 25 inches and the length of the wiper blade is 14 inches. Find the area wiped by the wiper blade. 2. A sprinkler on a golf green is set to spray water over a distance of 15 meters and to rotate through an
Determine whether the statement is true or false. Justify your answer. 1. An angle measure containing π must be in radian measure. 2. A measurement of 4 radians corresponds to two complete revolutions from the initial side to the terminal side of an angle. 3. The difference between the measures
Estimate the number of degrees in the angle.1.2. 3. 4.
1. When the radius of a circle increases and the magnitude of a central angle is held constant, how does the length of the intercepted arc change? Explain.2.3. A fan motor turns at a given angular speed. How does the speed of the tips of the blades change when a fan of greater diameter is installed
1. Is a degree or a radian the larger unit of measure? Explain. 2. Prove that the area of a circular sector of radius r with central angle θ is A = 1/2 θr2, where is measured in radians.
Determine the quadrant in which each angle lies. 1. (a) 130° (b) 285° 2. (a) 8.3° (b) 257° 30ˊ 3. (a) −132° 50ˊ (b) −336° 4. (a) −260° (b) −3.4°
Sketch each angle in standard position. 1. (a) 30° (b) 150° 2. (a) −270° (b) −120° 3. (a) 405° (b) −540° 4. (a) 750° (b) −600°
Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees.1. (a)(b) 2. (a) (b) 3. (a) Î¸ = ˆ’300° (b) Î¸ = 740° 4. (a) Î¸ = ˆ’520° (b) Î¸ = 230°
Convert each angle measure to decimal degree form. 1. (a) 85° 18ʹ 30ʺ (b) −330° 25ʺ 2. (a) 135° 36ʺ (b) −408° 16ʹ 20ʺ
Convert each angle measure to Do Mʹ Sʺ form. 1. (a) −345.12° (b) 0.45° 2. (a) 2.5° (b) −3.58°
Find (if possible) the complement and supplement of each angle. 1. (a) 18° (b) 85° 2. (a) 46° (b) 93° 3. (a) 24° (b) 126° 4. (a) 3° (b) 90°
Estimate the angle to the nearest one-half radian.1.2. 3. 4. 5. 6.
Determine the quadrant in which each angle lies. 1. (a) π/4 (b) 3π/4 2. (a) 11π/8 (b) 9π/8 3. (a) -π/5 (b) 7π/5
Sketch each angle in standard position. 1. (a) π/3 (b) - 2π/3 2. (a) - 7π/4 (b) 5π/2 3. (a) 17π/6 (b) -3 4. (a) 4 (b) 7 π
Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.1. (a)(b) 2. (a) (b)
Find (if possible) the complement and supplement of each angle. 1. (a) π/12 (b) 11π/12 2. (a) π/6 (b) 3π/4
Convert each degree measure to radian measure as a multiple of . Do not use a calculator. 1. (a) 30° (b) 45° 2. (a) 315° (b) 120° 3. (a) 20° (b) −60°
Convert each radian measure to degree measure. Do not use a calculator. 1. (a) 3π/2 (b) 7π/6 2. (a) -7π/12 (b) π/9 3. (a) 5π/12 (b) -7π/3
Convert the degree measure to radian measure. Round to three decimal places 1. 45° 2. 400° 3. 532.76° 4. −216.35° 5. −0.83°
Convert the radian measure to degree measure. Round to three decimal places 1. π/7 2. 5π/11 3. -2 4. -0.57
Find the length of the arc on a circle of radius r intercepted by a central angle Î¸.
Find the radian measure of the central angle.1.2. 3. 4.
Find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius r Arc Length s 1. 4 inches 18 inches 2. 14 feet 8 feet 3. 14.5 centimeters 0.2 meter 4. 700 meters 3.5 kilometers
Find the area of the sector of a circle of radius r and central angle θ Radius r Central Angle 1. 6 inches π/3 radians 2. 12 millimeters π/4 radian 3. 2.5 kilometers 225° 4. 1.4 miles 330°
Describe the error.1.2. A circle has a radius of 6 millimeters. The length of the arc intercepted by a central angle of 72° is
Find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other). City Latitude 1. Dallas, Texas 32° 47 9 N Omaha, Nebraska 41° 15 50 N 2. San Francisco, California 37° 47
1. The pointer on a voltmeter is 6centimeters in length (see figure). Find the number of degrees through which the pointer rotates when it moves 2.5 centimeters on the scale.2. The diameter of the drum on an electric hoist is 10 inches (see figure). Find the number of degrees through which the drum
1. A satellite in circular orbit 1250 kilometers above Earth makes one complete revolution every 110 minutes. Assuming that Earth is a sphere of radius 6400 kilometers, what is the linear speed (in kilometers per minute) of the satellite? 2. A car moves at a rate of 65 miles per hour, and the
The restaurant at the top of the Space Needle in Seattle, Washington, is circular and has a radius of 47.25 feet. The dining part of the restaurant revolves, making about one complete revolution every 48 minutes. A dinner party, seated at the edge of the revolving restaurant at 6:45 P.M. finishes
A popular theory that attempts to explain the ups and downs of everyday life states that each person has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by the sine functions below, where t is the number of days since birth. Physical (23 days): P = sin 2πt
(a) Use a graphing utility to graph the functions f (x) = 2 cos 2x + 3 sin 3x and g(x) = 2 cos 2x + 3 sin 4x. (b) Use the graphs from part (a) to find the period of each function. (c) Is the function h(x) = A cos ax + β sin βx, where a and β are positive integers, periodic? Explain.
Two trigonometric functions f and g have periods of 2, and their graphs intersect at x = 5.35. (a) Give one positive value of x less than 5.35 and one value of x greater than 5.35 at which the functions have the same value. (b) Determine one negative value of x at which the graphs intersect. (c) Is
When you stand in shallow water and look at an object below the surface of the water, the object will look farther away from you than it really is. This is because when light rays pass between air and water, the water refracts, or bends, the light rays. The index of refraction for water is 1.333.
Using calculus, it can be shown that the arctangent function can be approximated by the polynomial arctan x ≈ - x3 / 3 + x5 / 5 - x7 / 7 where x is in radians. (a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the
A bicycle's gear ratio is the number of times the free wheel turns for every one turn of the chain wheel (see figure). The table shows the numbers of teeth in the free wheel and the chain wheel for the first five gears of an 18-speed touring bicycle. The chain wheel completes one rotation for each
A model for the height h (in feet) of a Ferris wheel car is h = 50 + 50 sin 8t where t is the time (in minutes). (The Ferris wheel has a radius of 50 feet.) This model yields a height of 50 feet when t = 0. Alter the model so that the height of the car is 1 foot when t = 0.
The function f is periodic, with period c. So, f (t + c) = f (t). Determine whether each statement is true or false. Explain. a. f (t − 2c) = f (t) b. f(t + 1 / 2c) = f (1 / 2t) c. f (1/2 [t + c]) = f (1/2t) d. f (1/2[t + 4c]) = f (1 / 2t)
A surveyor in a helicopter is determining the width of an island, as shown in the figure.(a) What is the shortest distance d the helicopter must travel to land on the island? (b) What is the horizontal distance x the helicopter must travel before it is directly over the nearer end of the
Use the figure below.(a) Explain why Î”ABC, Î”ADE, and Î”AFG are similar triangles. (b) What does similarity imply about the ratios (c) Does the value of sin A depend on which triangle from part (a) is used to calculate it? Does the value of sin A change when you
Use a graphing utility to graph h, and use the graph to determine whether h is even, odd, or neither. a. h(x) = cos2 x b. h(x) = sin2 x
Given that f is an even function and g is an odd function, use the results of following Exercise to make a conjecture about each function h. a. h(x) = [f (x)]2 b. h(x) = [g(x)]2
The pressure P (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by P = 100 - 20 cos 8 πt / 3 where t is the time (in seconds). (a) Use a graphing utility to graph the model. (b) What is the period of the model? What does it represent in the context of the
Match each trigonometric function with its right triangle definition. (a) sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f ) Cotangent 1. Hypotenuse / adjacent 2. Adjacent / opposite 3. Hypotenuse / opposite 4. Adjacent / hypotenuse 5. Opposite / hypotenuse 6. Opposite / adjacent
In Exercises 1-4, find the exact values of the six trigonometric functions of the angle Î¸ for each of the two triangles. Explain why the function values are the same.1.2.
In Exercises 1-4, sketch a right triangle corresponding to the trigonometric function of the acute angle . Then find the exact values of the other five trigonometric functions of θ. 1. cos θ = 15 / 17 2. sin θ = 3 / 5 3. sec θ = 6 / 5 4. tan θ = 4 / 5
1. Relative to the acute angle , the three sides of a right triangle are the ________ side, the ________ side, and the ________. 2. Cofunctions of ________ angles are equal. 3. An angle of ________ represents the angle from the horizontal upward to an object, whereas an angle of________
In Exercises 1-4, construct an appropriate triangle to find the missing values. (0o ‰¤ Î¸ ‰¤ 90o, 0 ‰¤ Î¸ ‰¤ Ï€ / 2)Function Î¸ (deg) Î¸ (red) Function value1.2. 3. 4.
In Exercises 1-4, use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct mode.) 1. (a) sin 20.2° (b) csc 69.8° 2. (a) tan 12.75° (b) cot 12.75° 3. (a) cot π / 16 (b) tan π / 16 4. (a) sec π / 9 (b) cos π / 9
In Exercises 1-2, use the given function value(s) and the trigonometric identities to find the exact value of each indicated trigonometric function. 1. sin 60o = √3 / 2, cos 60o = 1/2 a. sin 30o b. cos 30o c. tan 360o d. cot 60o 2. sin 30o = 1/2, tan 30o = √3 / 3 a. csc 30o b. cot 60o c.
In Exercises 1-4, use trigonometric identities to transform the left side of the equation into the right side (0 < θ < π / 2). 1. tan θ cot θ = 1 2. cos θ sec θ = 1 3. tan α cos α = sin α 4. cot α sin α = cos α
In Exercises 1-4, find the exact values of the six trigonometric functions of the angle .1.2. 3. 4.
In Exercises 1-4, find each value of in degrees (0o < θ < 90o) and radians (0 < θ < π / 2) without using a calculator. 1. a. sin θ = 1 / 2 b. csc θ = 2 2. a. cos θ = √2 / 2 b. tan θ = 1 3. a. sec θ = 2 b. cot θ = 1 4. a. tan θ = √3 b. csc θ = √2
In Exercises 1-4, find the exact values of the indicated variables.1. Find x and y.2. Find x and r. 3. Find x and r. 4. Find x and r.
You are standing 45meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86thfloor (the observatory) is 82°. The total height of the building is another 123meters above the 86thfloor. What is the approximate height of the building? One of
A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is 132feet from the tower and 3feet from the tip of the shadow, the person's shadow starts to appear beyond the tower's shadow. (a) Draw a right triangle that
You are skiing down a mountain with a vertical height of 1250 feet. The distance from the top of the mountain to the base is 2500 feet. What is the angle of elevation from the base to the top of the mountain?
A biologist wants to know the width w of a river to properly set instruments for an experiment.From point A, the biologist walks downstream 100 feet and sights to point C (see figure). From this sighting, it is determined that Î¸ = 54°. How wide is the river?
A guy wire runs from the ground to a cell tower. The wire is attached to the cell tower 150 feet above the ground. The angle formed between the wire and the ground is 43° (see figure).(a) How long is the guy wire? (b) How far from the base of the tower is the guy wire anchored to the ground?
In traveling across flat land, you see a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9° (see figure). Approximate the height of the mountain.
A steel plate has the form of one-fourth of a circle with a radius of 60 centimeters. Two two-centimeter holes are drilled in the plate, positioned as shown in the figure. Find the coordinates of the center of each hole.
A tapered shaft has a diameter of 5centimeters at the small end and is 15centimeters long (see figure). The taper is 3°. Find the diameter d of the large end of the shaft.
Use a compass to sketch a quarter of a circle of radius 10 centimeters. Using a protractor, construct an angle of 20° in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. By actual measurement,
A 20-meter line is used to tether a helium-filled balloon. The line makes an angle of approximately 85° with the ground because of a breeze.(a) Draw a right triangle that gives a visual representation of the problem. Label the known quantities of the triangle and use a variable to represent the

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