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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

The growth of male Saanen goats can be approximated by the equationwhere W is the weight (in kilograms) after t weeks. Find the weight of a goat at 5 weeks, given that the weight at birth is 3.65 kg. Use Euler’s method with h = 1 week. dW dt -0.01189W + 0.92389W0.016,
In Exercises, assume a normal distribution.Customers at a certain pharmacy spend an average of $54.40, with a standard deviation of $13.50. What are the largest and smallest amounts spent by the middle 50% of these customers?
In an early article describing how people learn, the rate of change of the probability that a person performs a task correctly (p) with respect to time (t) is given bywhere k and m are constants related to the rate that the person learns the task. For this exercise, let m = 4 and k = 0.5.(a)
Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities. (Round probabilities to 4 decimal places.)(a) P(0 ≤ X ≤ 1) (b) P(X ≥ 1)(c) P(0 ≤ X ≤ 2) f(x) = = 20x4 9 20 9x5 if 0 ≤ x ≤ 1 if x > 1
The probability density function for the time required for a person to learn a certain task was given byfor 3 ≤ t ≤ 10 minutes. Find the median time for a person to learn the task. f(t) = 8 7(t - 2)²¹
Find the particular solution for each initial value problem. (Some solutions may give y implicitly.) dy dx 1 - 2x y + 3 y(0) = 16
The probability density function for the number of U.S. users of Facebook, a computer social network, was found to bewhere t was the number of years since birth on [14.4, 71.5]. Calculate the expected age of a Facebook user, as well as the standard deviation. S(t) = 1 1995 +32.66t
The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year.
Suppose that 0 < y0 < N. Let b = (N - y0)/y0 , and let y(x) = N/(1 + be-kx) for all x. Show the following.(a) 0 < y(x) < N for all x.(b) The lines y = 0 and y = N are horizontal asymptotes of the graph.(c) y(x) is an increasing function.(d) ((ln b)/k, N/2) is a point of inflection of
Suppose that 0 < N < y0 . Let b = (y0 - N)/y0 and letSee the figure. Show the following.(a) 0 < b < 1(b) The lines y = 0 and y = N are horizontal asymptotes of the graph.(c) The line x = (ln b)/k is a vertical asymptote of the graph.(d) y(x) is decreasing on ((ln b)/k, ∞) and on
A rumor spreads through a community of 500 people at the ratewhere N is the number of people who have heard the rumor at time t (in hours). Use Euler’s method with h = 0.5 hours to find the number who have heard the rumor after 3 hours, if only 2 people heard it initially.
Find the particular solution for each initial value problem. (Some solutions may give y implicitly.) dy √x- dx = xy; y(1) = 4
The lifetime of a printer costing $200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100
We studied Newton’s Law of Cooling, for whichwhere T is the temperature after t hours for a body with initial temperature of 98.6°F in a room with a constant temperature of 10°F. Use Euler’s method with h = 0.1 hours to approximate the temperature after 1 hour, and compare with the
Find the particular solution for each initial value problem. (Some solutions may give y implicitly.) et dy dx - ey = x² 1; y(0) = 42 -
The time between major earthquakes in the Southern California region is a random variable with probability density function defined bywhere t is measured in days. Find the expected value and the standard deviation of this probability density function. f(t) = 1 e 960 -t/960
Find the particular solution for each initial value problem. (Some solutions may give y implicitly.) dy dx + 3x²y = x²; y(0) = 2
The annual rainfall (in cm) in a remote Middle Eastern country is a random variable with probability density function defined by(a) Find the mean annual rainfall.(b) Find the standard deviation.(c) Find the probability of a year with rainfall less than 1 standard deviation below the mean. f(x)
Elasticity of demand was discussed in Chapter 6 on Applications of the Derivative, where it was defined asfor demand q and price p. Find the general demand equation q = f(p) for each elasticity function. E = p dq q dp e
Elasticity of demand was discussed in Chapter 6 on Applications of the Derivative, where it was defined asfor demand q and price p. Find the general demand equation q = f(p) for each elasticity function.E = 2 E = p dq q dp e
Sales (in thousands) of a certain product are declining at a rate proportional to the amount of sales, with a decay constant of 15% per year.(a) Write a differential equation to express the rate of sales decline.(b) Find a general solution to the equation in part (a).(c) How much time will pass
An insurance policy pays for a random loss X subject to a deductible of C, where 0 < C < 1. The loss amount is modeled as a continuous random variable with density functionGiven a random loss X, the probability that the insurance payment is less than 0.5 is equal to 0.64. Calculate C. Choose
The life span of a certain insect (in days) is uniformly distributed over the interval [20, 36].(a) What is the expected life of this insect?(b) Find the probability that one of these insects, randomly selected, lives longer than 30 days.
Find the particular solution for each initial value problem. (Some solutions may give y implicitly.) dy X- dx - - 2x²y + 3x² = 0; y(0) = 15
The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function ƒ, where ƒ(x) is proportional to (10 + x)-2. Calculate the probability that the lifetime of the machine part is less than 6. Choose one of the following.(a) 0.04 (b)
Find the particular solution for each initial value problem. (Some solutions may give y implicitly.) dy R2 dx + 4xy - e²¹³¹ = 0; y(1) = e²
We saw that the age of a randomly selected, alcohol-impaired driver in a fatal car crash is a random variable with probability density function given by(a) Find the expected age of a drunk driver in a fatal car crash.(b) Find the standard deviation of the distribution.(c) Find the probability that
The length of a petal on a certain flower varies from 1 cm to 4 cm and has a probability density function defined byFind the probabilities that the length of a randomly selected petal will be as follows.(a) Greater than or equal to 3 cm(b) Less than or equal to 2 cm(c) Between 2 cm and 3 cm
During the early days of the Internet, growth in the number of users worldwide could be approximated by an exponential function. The following table gives the number of worldwide users of the Internet.Use a calculator with exponential and logistic regression capabilities to complete the
Find all equilibrium points and determine their stability. dy dx (1 - e)(y-2)
A life insurance company invests $5000 in a bank account in order to fund a death benefit of $20,000. Growth in the investment over time can be modeled by the differential equationwhere i is the interest rate and A(t) is the amount invested at time t (in years). Calculate the interest rate that the
The clotting time of blood is a random variable t with values from 1 second to 20 seconds and probability density function defined byFind the following probabilities for a person selected at random.(a) The probability that the clotting time is between 1 and 5 seconds(b) The probability that the
The digestion time (in hours) of a fixed amount of food is exponentially distributed with a = 1.(a) Find the mean digestion time.(b) Find the probability that the digestion time is less than 30 minutes.
We saw that the age of a randomly selected driver in a fatal car crash is a random variable with probability density function given by ƒ(t) = 0.06049e-0.03211t for t in [16, 84].(a) Find the expected age of a driver in a fatal car crash.(b) Find the standard deviation of the distribution.(c) Find
The probability density function for the number of fatal traffic accidents was found to bewhere t is the number of hours since midnight on [0, 24]. Calculate the expected time of day at which a fatal accident will occur. S(t) = 1 100,716 (-2.0671³+ 78.9712-704.6t+4633),
Find all equilibrium points and determine their stability. dy dx = (y - 8) (y² - 1)
The average height of a member of a certain tribe of pygmies is 3.2 ft, with a standard deviation of 0.2 ft. If the heights are normally distributed, what are the largest and smallest heights of the middle 50% of this population?
The length of a telephone call (in minutes), t, for a certain town is a continuous random variable with probability density function defined by ƒ(t) = 3t-4 for t in [1, ∞). Find the expected length of a phone call.
H. R. Pulliam found that the time (in minutes) required by a predator to find a prey is a random variable that is exponentially distributed, with μ = 25.(a) According to this distribution, what is the longest time within which the predator will be 90% certain of finding a prey?(b) What is the
The mobility of an insect is an important part of its survival. Researchers have determined that the probability that a marked flea beetle, Phyllotreta cruciferae or Phyllotreta striolata, will be recaptured within a certain distance and time after release can be calculated from the probability
Researchers who study the abundance of the flour beetle, Tribolium castaneum, have developed a probability density function that can be used to estimate the abundance of the beetle in a population. The density function, which is a member of the gamma distribution, is ƒ(x) = 1.185 *
In Exercises, use Euler’s method to approximate the indicated function value for y = f(x) to 3 decimal places, using h = 0.2. dy dx = x + y²¹; y(0) = 1; find y(1)
The amount of a tracer dye injected into the bloodstream decreases exponentially, with a decay constant of 3% per minute. If 6 cc are present initially, how many cubic centimeters are present after 10 minutes?
The evapotranspiration index I is a measure of soil moisture. An article on 10- to 14-year-old heath vegetation described the rate of change of I with respect to W, the amount of water available, by the equation (a) According to the article, I has a value of 1 when W = 0. Solve the initial
The number of U.S. users (in millions) on Facebook, a computer social network, in 2018 is given in the table below.(a) Plot the data using the midpoint and number of users in each interval. What type of function appears to best match these data?(b) Use the regression feature on your graphing
In Exercises, use Euler’s method to approximate the indicated function value for y = f(x) to 3 decimal places, using h = 0.2. dy dx et + y; y(0) = 1; find y(0.6)
According to the National Center for Health Statistics, the life expectancy for a 55-year-old American female is 28.9 years. Assuming that, from age 55, the survival of American females follows an exponential distribution, determine the following probabilities.(a) The probability that a randomly
The time required for a person to learn a certain task is a random variable with probability density function defined byThe time required to learn the task is between 3 and 10 minutes. Find the probabilities that a randomly selected person will learn the task in the following lengths of time.(a)
According to the National Center for Health Statistics, life expectancy for a 70-year-old American male is 14.5 years. Assuming that, from age 70, the survival of American males follows an exponential distribution, determine the following probabilities.(a) The probability that a randomly selected
The time between major earthquakes in the Southern California region is a random variable with probability density functionwhere t is measured in days.(a) Find the probability that the time between a major earthquake and the next one is less than 365 days.(b) Find the probability that the time
An isolated fish population is limited to 4000 by the amount of food available. If there are now 320 fish and the population is growing with a growth constant of 2% a year, find the expected population at the end of 10 years.
Historians and biographers have collected evidence that suggests that President Andrew Jackson suffered from mercury poisoning. Researchers have measured the amount of mercury in samples of Jackson’s hair from 1815. The results of this experiment showed that Jackson had a mean mercury level of
A person’s weight depends both on the daily rate of energy consumed, say, C calories per day, and on the daily rate of energy expent, typically between 15 and 20 calories per pound per day. Using an average value of 17.5 calories per pound per day, a person weighing w pounds expends 17.5w
The cumulative number of deaths worldwide due to the H1N1 virus, or swine flu, at various days into the epidemic are listed below, where April 21, 2009, was day 1.Use a calculator with logistic regression capability to complete the following.(a) Plot the number of deaths y against the number of
Studies by the Federal Highway Administration suggest that speed limits on many thoroughfares are set arbitrarily, and are often artificially low. According to traffic engineers, the ideal speed limit should be the 85th percentile speed. This means the speed at or below which 85 percent of the
Let y = ƒ(x) and dy/dx = (x/2) + 4, with y(0) = 0. Use Euler’s method with h = 0.1 to approximate y(0.3) to 3 decimal places. Then solve the differential equation and find ƒ(0.30 to 3 decimal places. Also, find y3 - ƒ(x3).
The time between major earthquakes in the Taiwan region is a random variable with probability density functionwhere t is measured in days.(a) Find the probability that the time between a major earthquake and the next one is more than 1 year but less than 3 years.(b) Find the probability that the
The marginal sales (in hundreds of dollars) of a computer software company are given bywhere x is the number of months the company has been in business. Assume that sales were 0 initially.(a) Find the sales after 6 months.(b) Find the sales after 12 months. dy dx = бе.. 600.3x
The following table gives the historic and projected populations (in millions) of China and India.Use a calculator with logistic regression capability to do the following.(a) Letting t represent the years since 1950, plot the Chinese population on the y-axis against the year on the t-axis. Discuss
Studies by the Federal Highway Administration suggest that speed limits on many thoroughfares are set arbitrarily, and are often artificially low. According to traffic engineers, the ideal speed limit should be the 85th percentile speed. This means the speed at or below which 85 percent of the
Let y = ƒ(x) and dy/dx = 3 + 2y , with y(0) = 0. Construct a table for xi and yi like the one in Section 10.3, Example 2, for [0, 1], with h = 0.2. Then graph the polygonal approximation of the graph of y = ƒ(x).Section 10.3, Example 2Use Euler’s method to solve dy/dx = 3y + (1/2)ex, with y(0)
The frequency of alcohol-related traffic fatalities has dropped in recent years but is still high among young people. Based on data from the National Highway Traffic Safety Administration, the age of a randomly selected, alcohol-impaired driver in a fatal car crash is a random variable with
Over time, the number of original basic words in a language tends to decrease as words become obsolete or are replaced with new words. In 1950, C. Feng and M. Swadesh established that of the original 210 basic ancient Chinese words from 950 A.D., 167 were still being used. The proportion of words
The rate at which a new worker in a certain factory produces items is given bywhere y is the number of items produced by the worker per day, x is the number of days worked, and the maximum production per day is 150 items. Assume that the worker produces 15 items at the beginning of the first day on
What is the logistic equation? Why is it useful?
The National Highway Traffic Safety Administration records the time of day of fatal crashes. The table on the next page gives the time of day (in hours since midnight) and the frequency of fatal crashes in 2018.(a) Plot the data using the midpoint and the frequency. What type of function appears to
The rainfall (in inches) in a certain region is uniformly distributed over the interval [32, 44].(a) What is the expected number of inches of rainfall?(b) What is the probability that the rainfall will be between 38 and 40 in.?
Researchers have found that the probability P that a plant will grow to radius R can be described by the differential equationwhere D is the density of the plants in an area. Given the initial condition P(0) = 1, find a formula for P in terms of R. dP dR = -4TDRP²,
A recent report by the U.S. Census Bureau predicts that the U.S. Hispanic population will increase from 57.5 million in 2016 to 111.2 million in 2060. Assuming the unlimited growth model dy/dt = ky fits this population growth, express the population y as a function of the year t. Let 2000
Researchers have shown that the number of successive dry days that occur after a rainstorm for particular regions of Catalonia, Spain, is a random variable that is distributed exponentially with a mean of 8 days.(a) Find the probability that 10 or more successive dry days occur after a
We saw in a review exercise in Chapter 4 on Calculating the Derivative that driver fatality rates were highest for the youngest and oldest drivers. When adjusted for the number of miles driven by people in each age group, the number of drivers in fatal crashes goes down with age, and the age of a
The following table gives the population of the world at various times over the past two centuries, plus projections for this century.Use a calculator with logistic regression capability to complete the following.(a) Use the logistic regression function on your calculator to determine the logistic
The length of a telephone call (in minutes), t, for a certain town is a continuous random variable with probability density function defined by ƒ(t) = 3t-4, for t in [1, ∞). Find the probabilities for the following situations.(a) The call lasts between 1 and 2 minutes.(b) The call lasts between
The report also predicted that the U.S. Asian population would increase from 18.3 million in 2016 to 36.8 million in 2060. Repeat Exercise 55 using these data.Exercise 55A recent report by the U.S. Census Bureau predicts that the U.S. Hispanic population will increase from 57.5 million in 2016 to
The proportion of the times (in days) between major earthquakes in the north-south seismic belt of China is a random variable that is exponentially distributed, with a = 1/609.5.(a) Find the expected number of days and the standard deviation between major earthquakes for this region.(b) Find the
A retirement savings account contains $300,000 and earns 5% interest compounded continuously. The retiree makes continuous withdrawals of $20,000 per year.(a) Write a differential equation to describe the situation.(b) How much will be left in the account after 10 years?
The time between goals (in minutes) for the Wolves soccer team in the English Premier League during a recent season can be approximated by an exponential distribution with a = 1/90.(a) The Wolves scored their first goal of the season 71 minutes into their first game. Find the probability that the
In Exercise 57, approximately how long will it take to use up the account?Exercise 57A retirement savings account contains $300,000 and earns 5% interest compounded continuously. The retiree makes continuous withdrawals of $20,000 per year.
The margin of victory over the point spread (defined as the number of points scored by the favored team minus the number of points scored by the underdog minus the point spread, which is the difference between the previous two, as predicted by oddsmakers) in National Football League games has been
After use of an experimental insecticide, the rate of decline of an insect population iswhere t is the number of hours after the insecticide is applied. Assume that there were 50 insects initially.(a) How many are left after 24 hours?(b) How long will it take for the entire population to die?
A competing model of health care costs to the model given in the previous exercise is the uniform price elasticity model. Using the definition of elasticity of demand from Section 6.3, this model assumeswhere k is a constant.(a) Solve the equation in part (a). What happens as p approaches 0?(b) To
A company has found that the rate at which a person new to the assembly line produces items iswhere x is the number of days the person has worked on the line. How many items can a new worker be expected to produce on the eighth day if he produces none when x = 0? dy dx || 7.5e-0.3y.
Find an equation relating x to y given the following equations, which describe the interaction of two competing species and their growth rates.Find the values of x and y for which both growth rates are 0. dx dt dy dt = 0.2x - 0.5xy = -0.3y + 0.4xy
A population of mites grows at a rate proportional to the number present, y. If the growth constant is 10% and 120 mites are present at time t = 0 (in weeks), find the number present after 6 weeks.
Suppose the rate at which a rumor spreads—that is, the number of people who have heard the rumor over a period of time—increases with the number of people who have heard it. If y is the number of people who have heard the rumor, thenwhere t is the time in days.(a) If y is 1 when t = 0, and y is
The air in a meeting room of 15,000 ft3 has a smoke content of 20 parts per million (ppm). An air conditioner is turned on, which brings fresh air (with no smoke) into the room at a rate of 1200 ft3 per minute and forces the smoky air out at the same rate. How long will it take to reduce the smoke
In Exercise 62, how long will it take to reduce the smoke content to 10 ppm if smokers in the room are adding smoke at the rate of 5 ppm per minute?Exercise 62The air in a meeting room of 15,000 ft3 has a smoke content of 20 parts per million (ppm). An air conditioner is turned on, which brings
One morning, snow began to fall at a heavy and constant rate. A snowplow started out at 8:00 a.m. At 9:00 a.m. it had traveled 2 miles. By 10:00 a.m. it had traveled 3 miles. Assuming that the snowplow removes a constant volume of snow per hour, determine the time at which it started snowing.
A small, isolated mountain community with a population of 700 is visited by an outsider who carries influenza. After 6 weeks, 300 people are uninfected.(a) Write an equation for the number of people who remain uninfected at time t (in weeks).(b) Find the number still uninfected after 7 weeks.(c)
The amount of a radioactive substance decreases exponentially, with a decay constant of 3% per month.(a) Write a differential equation to express the rate of change.(b) Find a general solution to the differential equation from part (a).(c) If there are 75 g at the start of the decay process, find a
LetIf y is y1, y2, and y3 at times t1, t2, and t3 = 2t2 - t1 (that is, at three equally spaced times), then prove that y N 1 + bekt
Newton’s law of cooling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and TM is the (constant) temperature of the surrounding
The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. Choose one of the following.(a) 0.07 (b) 0.29 (c) 0.38 (d)
A machine has a useful life of 4 to 9 years, and its life (in years) has a probability density function defined byFind the probabilities that the useful life of such a machine selected at random will be the following.(a) Longer than 6 years(b) Less than 5 years(c) Between 4 and 7 years(d) Find the
Use the methods of this section to find the volume of a cylinder with height h and radius r.
Find the relative maxima or minima in Exercises.Maximum of ƒ(x, y) = 12xy - x2 - 3y2, subject to x + y = 16
Evaluate dz using the given information. N 11 y² + 3x 2 - X ; x = 4, y 4, y = -4, dx = 0.01, dy = 0.03
In the following table of U.S. Census figures, y is the population in millions.Use Exercise 65 and the table on the previous page to find the following.(a) Find N using the years 1800, 1850, and 1900.(b) Find N using the years 1850, 1900, and 1950.(c) Find N using the years 1870, 1920, and
Let t = 0 correspond to 1790, and let every decade correspond to an increase in t of 1. Use a calculator with logistic regression capability to complete the following. Use the table from Exercise 66.(a) Plot the data points. Do the points suggest that a logistic function is appropriate here?(b) Use
Newton’s law of cooling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and TM is the (constant) temperature of the surrounding

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