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College Algebra With Modeling And Visualization 6th Edition Gary Rockswold - Solutions

The salinity of the oceans changes with latitude and with depth. In the tropics, the salinity increases on the surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation and rainfall causes the salinity to be less on the surface than at lower depths. The
The life span of a sample of sparrows was studied. The equationcalculates the numbers of years y required for x percent of the sparrows to die, where 0 ≤ x ≤ 95.(a) Find y when x = 40. Interpret your answer. (b) Find x when y = 1.5. Interpret your answer. 2 log (100 0.37
Use the tables to evaluate the following. 0 f(x) 8 X x 2 2 0 2 8 g(x) 4 4 6 0 6 4 0 6 2 8 4 8 6
Chloro- fluorocarbons (CFCs) are gases that might increase the greenhouse effect. The following table lists future concentrations of CFC-12 in parts per billion (ppb) if current trends continue.(a) Find values for C and a so that f(x) = Cax models these data, where x is years after 2000. (b)
The formula C(x) = 280ln(x + 1) + 1925 models the number of calories consumed daily by a person owning x acres of land in a developing country. Estimate the number of acres owned for which average intake is 2300 calories per day.
Use transformations to graph y = g(x). Give the equation of any asymptotes. g(x) = 3logx
Use the tables to evaluate the following. 0 f(x) 8 X x 2 2 0 2 8 g(x) 4 4 6 0 6 4 0 6 2 8 4 8 6
Use transformations to graph y = g(x). Give the equation of any asymptotes. g(x) = 2log₂x
A pollutant in a river has an initial concentration of 3 parts per million and degrades at a rate of 1.5% per year. Approximate its concentration after 20 years.
The graph of f computes the balance in a savings account after x years. Estimate each expression. Interpret what f-1(x) computes. (a) f(1) Balance (dollars) 180 170 160 150 140 130 120 110 100 90 (b) f¹(110) (c) f¹(160) 01 2 y = f(x) 3 Years 4 5 6
The population of Arizona was 6.6 million in 2010 and growing continuously at a 1.44% rate. Assuming this trend continued, estimate the population of Arizona in 2016.
The graph off at the top of the next column computes the Celsius temperature of a pan of water after x minutes. Estimate each expression. Interpret what the expression f-1(x) computes. (a) f(4) (b) ƒ-¹(90) (c) ƒ-¹(80)
The table lists the concentration of a sample of E. coli bacteria B (in billions per liter) after x hours.(a) Find values for C and a so that f(x) = Cax models these data. (b) Estimate the bacteria concentration after 6.2 hours. x 0 B 0.5 3 6.2 5 5 33.3 8 414
Use transformations to graph y = g(x). Give the equation of any asymptotes g(x) = log₂ (-x)
The surface area A of a balloon with radius r is given by 4(r) = 4πr2. Suppose that the radius of the balloon increases from r to r + h, where his a small positive number. (a) Find A(r + h) A(r). Interpret your answer. (b) Evaluate your expression in part (a) for r = 3 and h = 0.1, and
The concentration of bacteria in a sample can be modeled by B(t) = B0ekt, where t is in hours and B is the concentration in billions of bacteria per liter. (a) If the concentration increases by 15% in 6 hours, find k. (b) If B0 1.2, find B after 8.2 hours.(c) By what percentage does the
Use the graph to evaluate the expression. I 3 y = f(x) N 3 (a) f(-1) (b) f¹(-2) (c) f¹(0) (d) (f¹f)(3)
Suppose $1500 is deposited into an IRA with an interest rate of 6%, compounded continuously. How much money will there be after 30 years?
Use the graph to evaluate the expression. y=f(x) 321 1 3 (a) f(1) (b) f¹(1) (c) ƒ-¹(4) (d) (ƒ•ƒ-¹)(2.5)
The voltage in a circuit can be modeled by V(1) = V0ekt, where t is in milliseconds. (a) If the voltage decreases by 85% in 5 milliseconds, find k. (b) If V0 = 4.5 volts, find V after 2.3 milliseconds. (c) By what percentage does the voltage decrease each millisecond?
Use transformations to graph y = g(x). Give the equation of any asymptotes DO (x) = -log₂.x
(a) Approximate the number of E. coli after 3 hours. (b) Estimate graphically the elapsed time when there are 10 million bacteria per milliliter.
Use the graph to evaluate the expression. y*f(x) 24 8 (a) f(4) (b) f¹(0) (c) ƒ-¹(6) (d) (ƒ•ƒ¹)(4)
Use transformations to graph y = g(x). Give the equation of any asymptotes g(x) = 2 + In (x - 1)
A marble is dropped into a lake, resulting in a circular wave whose radius increases at a rate of 6 inches per second. Write a formula for C that gives the circumference of the circular wave in inches after / seconds.
Methane is a greenhouse gas that lets sunlight into the atmosphere but blocks heat from escaping Earth's atmosphere. In the table, f(x) models the predicted methane emissions in millions of tons produced by developed countries. The function g(x) models the same emissions for developing
Ecologists studied the spacing between individual trees in a forest in British Columbia. The probability, or likelihood, that there is at least one tree located in a circle with a radius of x feet can be estimated by P(x) = 1 - e-1144x. For example, P(7) ≈ 0.55 means that if a person picks a
Energy of a Falling Object A ball with mass m is dropped from an initial height of h0 and lands with a final velocity of vf. The kinetic energy of the ball is K(v) = 1/2mv2, where v is its velocity, and the potential energy of the ball is P(h0) = mgh, where h is its height and g is a
The figure shows graphs of the functions f and g that model methane emissions. Use these graphs to sketch a graph of the function h. Methane emissions (million tons) 1990 2010 Year y = f(x) y = g(x) 2030
For the given annual interest rate r, estimate the time for P dollars to double.P = $1000, r = 8.5% compounded quarterly
Sometimes after a patient takes a drug, the amount of medication A in the bloodstream can be modeled by A = A0e-rt, where A0 is the initial concentration in milligrams per liter, r is the hourly percentage decrease (in decimal form) of the drug in the bloodstream, and t is the elapsed time in
Use transformations to graph y = g(x). Give the equation of any asymptotes g(x)= ln(x + 1)-1
Cars arrive randomly at an intersection with an average rate of 50 cars per hour. The likelihood, or probability, that at least one car will enter the intersection within a period of x minutes can be estimated by P(x) = 1 - e-5x/6 (a) Find the likelihood that at least one car enters the
For the given annual interest rate r, estimate the time for P dollars to double.P = $750, r = 2% compounded continuously
Write a function A that gives the area contained inside the circular wave in square inches after / seconds.
Graph f and state its domain. f(x) = log(x + 1)
Suppose that P dollars is deposited in a savings account paying 3% interest compounded continuously. After t years, the account will contain A (t) = Pe0.03t dollars. (a) Solve A (t) = b for the given values of P and b. (b) Interpret your results.P = 500 and b = 750
The surface area of a cone (excluding the bottom) is given by S = πr √r2 + h2, where r is its radius and his its height, as shown in the figure. If the height is twice the radius, write a formula for S in terms of r. h
Suppose that P dollars is deposited in a savings account paying 3% interest compounded continuously. After t years, the account will contain A (t) = Pe0.03t dollars. (a) Solve A (t) = b for the given values of P and b. (b) Interpret your results.P = 1000 and b = 2000
The percentage P of radio-active carbon-14 remaining in a fossil after / years is given by P = 100(1/2)t/5700 Suppose a fossil contains 35% of the carbon-14 that the organism contained when it was alive. Estimate the age of the fossil.
The half-life for a link on Twitter is 2.8 hours. Write an exponential function T that gives the percentage of engagements remaining on a typical Twitter link after 1 hours. Estimate this percentage after 5.5 hours.
The amount A of radium in milligrams remaining in a sample after 7 years is given by A(1) = 0.02(1/2)t/1600. How many years will it take for the radium to decay to 0.004 milligram?
Use the tables to evaluate the following. x012 3 5 f(x) 1 1 2 0 2 1 x-1 3 4 3 4 4 2 4 5
The global sea level could rise due to partial melting of the polar ice caps. The table represents a function R that models this expected rise in sea level in centimeters for the year t. (a) Is R a one-to-one function? Explain. (b) Use R(t) to find a table for R-1(1). Interpret R-1. 1(yr)
Wind speed typically varies in the first 20 meters above the ground. For a particular day, let the formula f(x) = 1.2 ln x + 2.3 compute the wind speed in meters per second at a height x meters above the ground for x ≥ 1. (a) Find the wind speed at a height of 5 meters. (b) Graph f in the
Graph y = f(x), y = f-1(x), and y = x in a square viewing rectangle such as [-4.7, 4.7, 1] by [-3.1, 3.1, 1]. f(x) = 3x - 1
Graph y = f(x) and y = x. Then graph y = f-1(x). f(x) = √x+1
Hurricanes are some of the largest storms on Earth. They are very low pressure areas with diameters of over 500 miles. The barometric air pressure in inches of mercury at a distance of x miles from the eye of a severe hurricane is modeled by the formula f(x) = 0.48 In (x + 1) + 27. (a) Find
Graph y = f(x), y = f-1(x), and y = x in a square viewing rectangle such as [-4.7, 4.7, 1] by [-3.1, 3.1, 1]. 1 = x ² = (x)/ I
Graph y = f(x), y = f-1(x), and y = x in a square viewing rectangle such as [-4.7, 4.7, 1] by [-3.1, 3.1, 1]. f(x) 3-x 2
Graph y = f(x), y = f-1(x), and y = x in a square viewing rectangle such as [-4.7, 4.7, 1] by [-3.1, 3.1, 1]. I-XA = (x)f
The tables represent a function F that converts yards to feet and a function Y that converts miles to yards. Evaluate each expression and interpret the results. x(yd) 1760 F(x) (ft) 3520 5280 7040 8800 5280 10,560 15,840 21,120 26,400 x(mi) Y(x) (yd) 1 1760 2 3520 (a) (Fo Y)(2) (b) F-¹(26,400) (c)
The table lists the number of species of birds on islands of various sizes. Find values for a and b so that f(x) = a + blogx models these data. Estimate the size of an island that might have 16 species of birds. (Let f(1) = 7 and find a. Then let f(10) = 11 and find b.) Area (km²) Species of
Use the graph to evaluate the expression. y-f(x) 23 (a) f(1) (b) f¹(-1) (c) ƒ-¹(3) (d) (ff-¹)(1)
A pot of boiling water with a temperature of 100°C is set in a room with a temperature of 20°C. The temperature T of the water after x hours is given by T(x) = 20 + 80e-x. (a) Estimate the temperature of the water after 1 hour. (b) How long did it take the water to cool to 60°C?
The table lists the number of bacteria y in millions after an elapsed time of x days.(a) Find values for C and a so that f(x) = Cax models the data. (b) Estimate when there were 16 million bacteria. 0 3 1 6 2 12 3 24 4 48
The tables represent a function C that converts tablespoons to cups and a function Q that converts cups to quarts. Evaluate each expression and interpret the results. x(tbsp) C(x) (c) x(c) Q(x) (qt) 32 2 64 4 2 4 0.5 1 (a) (QC)(96) (b) Q¹(2) (c) (C-¹0¹)(1.5) 96 6 6 1.5 128 8 8 2
The table lists the number of types of insects found in wooded regions having various acreages. Find values for a and b so that f(x) = a + blog.x models these data. Use to esti- mate an acreage that might have 1200 types of insects. Area (acres) Insect Types 10 500 100 800 1000 10,000 1100 1400
Explain how to solve the equation blogax = k symbolically for x. Demonstrate your method.
The growth of an investment is shown in the table.(a) Find values for C and a so that f(x) = Cax models the data. (b) Estimate when the account contained $2000. x (years) y (dollars) 0 0 5 10 15 20 100 300 900 2700 8100
Discuss the domain and range of an exponential function f. Is f one-to-one? Explain.
The formula W = 25/7h - 800 weight approximates the recommended minimum for a person h inches tall, where 62 ≤ h ≤ 76. (a) What is the recommended minimum weight for someone 70 inches tall? (b) Does W represent a one-to-one function? (c) Find a formula for the inverse. (d)
A can of soda with a temperature of 5°C is set in a room with a temperature of 20°C. The temperature T of the soda after x minutes is given by T(x) = 20 - 15(10)-0.05x.(a) Estimate the temperature of the soda after 5 minutes. (b) After how many minutes was the temperature of the soda 15°C?
The population of Nevada in millions is given by P(x) = 2.7e0.014x, where x = 0 corresponds to 2010. (a) Determine symbolically the year when the population of Nevada might be 3 million. (b) Solve part (a) graphically.
Cars arrive randomly at an intersection with an average rate of 20 cars per hour. The likelihood, or probability, that no car enters the intersection within a period of x minutes can be estimated by f(x) = e-x/3.(a) What is the likelihood that no car enters the intersection during a 5-minute
The formula F = 9/5C + 32 converts a Celsius temperature to Fahrenheit temperature. (a) Find a formula for the inverse. (b) Normally we interchange x and y to find the inverse function. Does it makes sense to inter- change F and C in part (a) of this exercise? Explain. (c) What
The volume V of a sphere with radius r is given by V = 4/3πr3. (a) Does V represent a one-to-one function? (b) What does the inverse of V compute?(c) Find a formula for the inverse. (d) Normally we interchange x and y to find the inverse function. Does it make sense to interchange V
Graph f and state its domain. f(x) = log(x-3)
(a) Find formulas for F(x), Y(x), and (F ° Y)(x). (b) Find a formula for (Y-1 ° F-1)(x). What does this function compute?
The formula T(x) = x3/2 calculates the time in years that it takes a planet to orbit the sun if the planet is x times farther from the sun than Earth is. (a) Find the inverse of T. (b) What does the inverse of T calculate?
Describe the relationship among exponential functions and logarithmic functions. Explain why logarithms are needed to solve exponential equations.
The half-life for a link on StumbleUpon is 400 hours. Write an exponential function S that gives the percentage of engagements remaining on a typical StumbleUpon link after 1 hours. Estimate this percentage after 250 hours.
Give verbal, numerical, graphical, and symbolic representations of a base-5 logarithmic function.
At a distance of 3 feet, professional golfers make about 95% of their putts. For each additional foot of distance, this percentage decreases by a factor of 0.9. (a) Write an exponential function P that gives the percentage of putts pros make at a distance of x feet beyond a 3-foot
Use the graph of y = f(x) to sketch a graph of y = f-1(x). -3 -1 3 (-2,-4)-3 1 y = f(x) (1.2). 3 X
The probability that a car will enter an intersection within a period of x minutes is given by P(x) = 1 -e-0.5x Determine symbolically the elapsed time x when there is a 50-50 chance that a car has entered the intersection. (Solve P(x) = 0.5.)
The volume V of a sphere with radius r is given by V = 4/3πr3, and the surface area S is given by S = 4πr2. Show that V=(4/2)π (S/4π)3/2
Graph f and state its domain. f(x) = ln(-x)
Cars arrive randomly at an intersection with an average traffic volume of one car per minute. The likelihood, or probability, that at least one car enters the intersection during a period of x minutes can be estimated by f(x) = 1 = e-x (a) What is the probability that at least one car enters
Use the graph of y = f(x) to sketch a graph of y = f-1(x). 7 5 lad (3.9) y=f(x) (2,4) 1 (1.1) 01 3 5 7 79
Impurities in water are frequently removed using filters. Suppose that a 1-inch filter allows 10% of the impurities to pass through it. The other 90% is trapped in the filter. (a) Find a formula in the form f(x) = 100ax that calculates the percentage of impurities passing through x inches of
Graph f and state its domain. f(x) = In (x² + 1)
Use the graph of y = f(x) to sketch a graph of y = f-1(x). 3
Suppose that the concentration of a bacteria sample is 100,000 bacteria per milliliter. If the concentration doubles every 2 hours, how long will it take for the concentration to reach 350,000 bacteria per milliliter?
Use the graph of y = f(x) to sketch a graph of y = f-1(x). (-2.4) (-4.0) -2 6 2 75 J = f(x) 2 6
Use the graph of y = f(x) to sketch a graph of y = f-1(x). -3 -1 3y = f(x) 32 _77 3 123 X
A fossil contains 10% of the carbon-14 that the organism contained when it was alive. Graphically estimate its age.
The concentration of a sample of bacteria (in millions per milliliter) after x days is given by C(x) = 4(2x). Determine both the initial concentration and the number of days before the concentration reaches 20 million per milliliter.
Suppose that the concentration of a bacteria sample is 50,000 bacteria per milliliter. If the concentration triples in 4 days, how long will it take for the concentration to reach 85,000 bacteria per milliliter?
A fossil contains 20% of the carbon-14 that the organism contained when it was alive. Estimate its age.
The density of an insect population (in thousands per acre) after x weeks is given by D(x) = 600(2/3)x. Determine both initial density and the number of weeks before the density reaches 100 thousand per acre.
Suppose that $2000 is deposited in an account and the balance increases to $2300 after 4 years. How long will it take for the account to grow to $3200? Assume continuous compounding.
The half-life of radium-226 is about 1600 years. After 3000 years, what percentage P of a sample of radium remains?
Sound levels in decibels (dB) can be calculated by D(x) = 10log (1016x), where x is the intensity of the sound in watts per square meter. The human ear begins to hurt when the intensity reaches x = 10-4. Find how many decibels this represents.
Suppose that a 0.05-gram sample of a radioactive substance decays to 0.04 gram in 20 days. How long will it take for the sample to decay to 0.025 gram?
Use the graph of y = f(x) to sketch a graph of y = f-1(x). 32 y = f(x) -3-2-1 1 3
Radioactive strontium-90 has a half-life of about 28 years and some-times contaminates the soil. After 50 years, what percentage of a sample of radioactive strontium would remain?
If the intensity increases by a factor of 10, find the increase in decibels.
The concentration of a drug in a patient's bloodstream after t hours is modeled by the formula C(t) = 11(0.72)t, where C is measured in milligrams per liter.(a) What is the initial concentration of the drug? (b) How long does it take for the concentration to decrease to 50% of its initial
Graph y = f(x) and y = x. Then graph y = f-1(x). f(x) = 2x - 1

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