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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
An economist studying the demand for a particular commodity gathers the data in the accompanying table, which lists the number of units q (in thousands) of the commodity that will be demanded (sold) at a price of p dollars per unit.Use this information together with Simpson’s rule to estimate the
Suppose the time X a customer must spend waiting in line at a certain bank is a random variable that is exponentially distributed with density functionwhere x is the number of minutes a randomly selected customer spends waiting in line.a. Find the probability that a customer will have to stand in
It is estimated that t years from now an apartment complex will be generating profit for its owner at the rate of f(t) = 80,000 + 500t dollars per year, dispensed continuously. If the profit is generated in perpetuity and the prevailing annual interest rate remains fixed at 5% compounded
The capitalized cost of an asset is the sum of the original cost of the asset and the present value of maintaining the asset. Suppose a company is considering the purchase of two different machines. Machine 1 costs $10,000 and t years from now will cost M1(t) = 1,000(1 + 0.06t) dollars per year to
In Exercises 35 through 38, integrate the given probability density functions to find the indicated probabilities.a. P(0 ≤ X ≤ 3)b. P(1 ≤ X ≤ 2) f(x) = 0 (3x) if 0≤x≤3 otherwise
In Exercises 35 through 38, integrate the given probability density functions to find the indicated probabilities.a. P(X ≥ 0) b. P(1 ≤ X ≤ 4)c. P(X ≥ 5) f(x) = (0.2e 0.2e-0.2x to if x ≥ 0 if x < 0
Use Simpson’s rule with n = 6 to estimate the volume of the solid generated by rotating the region under the curve y = ln x between x = 1 and x = 2 about the x axis.
In t years, an investment will be generating f(t) = A + Bt dollars per year, where A and B are constants. Suppose the income is generated in perpetuity, with a fixed annual interest rate r compounded continuously. Show that the present value of this investment isdollars. A r + B 2 r
The management of a national chain of fast-food restaurants is selling a 5-year franchise to operate its newest outlet in Tulare, California. Past experience in similar localities suggests that t years from now, the franchise will be generating profit at the rate of f(t) = 12,000√t dollars per
The management of a national chain of fast-food outlets is selling a permanent franchise in Seattle, Washington. Past experience suggests that t years from now, the franchise will be generating profit in perpetuity at the rate of f(t) = 100,000 + 900t dollars per year. If the prevailing interest
An investment generates income continuously at the rate of f(t) = √t thousand dollars per year at time t (years). If the prevailing rate of interest is 6% per year compounded continuously, use the trapezoidal rule with n = 5 to estimate the future value of the investment over a 10-year term.
In Exercises 35 through 38, integrate the given probability density functions to find the indicated probabilities.a. P(X > 0) b. P(1 ≤ X ≤ 9)c. P(X ≥ 3) f(x) = 5 (x + 5)² if x ≥ 0 if x < 0
Jacob Lawrence is an economist who models the demand for a particular commodity by the functionwhere q hundred units are sold when the price is p dollars per unit. He decides to use Simpson’s rule with n = 6 to estimate the consumers’ surplus for the commodity when 500 units (q0 = 5) are
The duration of telephone calls in a certain city is measured by a random variable X with probability density functionwhere x denotes the duration (in minutes) of a randomly selected call.a. What is the probability that a randomly selected call lasts between 2 and 3 minutes?b. What percentage of
Marco has a small investment providing a variable income stream that is deposited continuously into an account earning interest at an annual rate of 4% compounded continuously. He spot-checks the monthly flow rate of the investment on the first day of every other month for a 1-year period,
The life span of a particular brand of food processor is measured by a random variable X with probability density functionwhere x denotes the life span (in months) of a randomly selected processor.a. Verify that f(x) is a probability density function by showing thatb. The expected value of X is
On the first day of each month, the manager of a small company estimates the rate at which profit is expected to increase during that month. The results are listed in the accompanying table for the first 6 months of the year, where P´(t) is the rate of profit growth in thousands of dollars per
An economist studying the supply for a particular commodity gathers the data in the accompanying table, which lists the number of units q (in thousands) of the commodity that will be supplied to the market by producers at a price of p dollars per unit. Use this information together with the
A group of patients with a potentially fatal disease has been treated with an experimental drug. Assume the survival time X for a patient receiving the drug is a random variable exponentially distributed with density functionwhere x is the number of years a patient survives after first receiving
The manager of an electronics firm estimates that the proportion of components that last longer than t months is given by the improper integralWhich is larger, the proportion of components that last longer than 5 years (60 months) or the proportion that fail in less than 10 years? S -0.008x
The reliability function r(x) of an electronic component is the probability that the component continues to work for more than x days. Suppose that the working life of a component is measured by a random variable X with probability density function f(x).a. Find a formula for r(x) in terms of
An economist studying the demand for a particular commodity gathers the data in the accompanying table, which lists the number of units q (in thousands) of the commodity that will be demanded (sold) at a price of p dollars per unit. Use this information together with Simpson’s rule to estimate
A county mental health clinic has just opened. The clinic initially accepts 300 people for treatment and plans to accept new patients at the rate of 10 per month. Let f(t) denote the fraction of people receiving treatment continuously for at least t days. For the first 60 days, records are kept and
In Exercises 39 through 42, solve the given initial value problem for y = f(x). Note that Exercises 41 and 42 involve separable differential equations. dy dx xy √x + 1 where y = 1 when x = 0
A certain traffic light remains red for 45 seconds at a time. You arrive (at random) at the light and find it red. Use an appropriate uniform density function to find:a. The probability that the light will turn green within 15 seconds.b. The probability that the light will turn green between 5 and
An investment will generate income continuously at the constant rate of Q dollars per year in perpetuity. Assuming a fixed annual interest rate r compounded continuously, use an improper integral to show that the present value of the investment is Q/r dollars.
In Exercises 39 through 42, solve the given initial value problem for y = f(x). Note that Exercises 41 and 42 involve separable differential equations.dy/dx = xe−2x, where y = 0 when x = 0
A sociologist studying the distribution of income in an industrial society compiles the data displayed in the accompanying table, where L(x) denotes the fraction of the society’s total wealth earned by the lowest-paid 100x% of the wage-earners in the society. Use this information together with
During the morning rush hour, commuter trains run every 20 minutes from the station near Quon’s home into the city center. He arrives (at random) at the station and finds no train at the platform. Assuming that the trains are running on schedule, use an appropriate uniform density function to
Demographic studies conducted in a certain city indicate that the fraction of the residents that will remain in the city for at least t years is f(t) = e−t/20. The current population of the city is 100,000, and it is estimated that t years from now, new people will be arriving at the rate of 100t
The time interval between the arrivals of successive planes at a certain airport is measured by a random variable X with probability density functionwhere x is the time (in minutes) between the arrivals of a randomly selected pair of successive planes.a. What is the probability that two successive
Suppose the length of time that it takes a laboratory rat to traverse a certain maze is measured by a random variable X that is exponentially distributed with density functionwhere x is the number of minutes a randomly selected rat spends in the maze.a. Find the probability that a randomly selected
In Exercises 39 through 42, solve the given initial value problem for y = f(x). Note that Exercises 41 and 42 involve separable differential equations.dy/dx = x2 ln x, where y = 0 when x = 1
Demographic studies conducted in a certain city indicate that the fraction of the residents that will remain in the city for at least t years is f(t) = e−t/20. The current population of the city is 200,000, and it is estimated that new residents will be arriving at the rate of 100 people per
Let X be a random variable that measures the age of a randomly selected cell in a particular population. Suppose X is distributed exponentially with a probability density function of the formwhere x is the age of a randomly selected cell (in days) and k is a positive constant. Experiments indicate
The fraction of patients who will still be receiving treatment at a certain health clinic t months after their initial visit is f(t) = e−t/20. If the clinic accepts new patients at the rate of 10 per month, approximately how many patients will be receiving treatment at the clinic in the long run?
The proportion P of susceptible people who are infected t weeks after the outbreak of an epidemic is given by the integralwhere a and b are parameters that depend on the disease and C is a constant. Assuming that all susceptible people are eventually infected, find C (in terms of a and b).
In Exercises 39 through 42, solve the given initial value problem for y = f(x). Note that Exercises 41 and 42 involve separable differential equations.dy/dx = xyex/2, where y = 1 when x = 0
When excavation is being done for roads or buildings, the contractor will often pay nearby landowners to dump the excavated earth on their land, reducing the cost of hauling it away. One landowner, Dana Mays, is offered $0.20 per cubic foot to have dirt dumped to a depth of 2 feet over an oddly
The publishers of a national magazine have found that the fraction of subscribers retained for at least t years is f(t) = e−t/10. Currently, the magazine has 20,000 subscribers and it is estimated that new subscriptions will be sold at the rate of 1,000 per year. Approximately how many
Suppose the length of time that it takes a laboratory rat to traverse a certain maze is measured by a random variable X that is exponentially distributed with density functionwhere x is the number of minutes a randomly selected rat spends in the maze.a. Find the probability that a randomly selected
Marta receives 5 units of a certain drug per hour intravenously. The drug is eliminated exponentially, so that the fraction that remains in Marta’s body for t hours is f(t) = e−t/10. If the treatment is continued indefinitely, approximately how many units of the drug will be in Marta’s body
An industrial plant spills pollutant into a river. The pollutant spreads out as it is carried downstream by the current of the river, and 3 hours later, the spill forms the pattern shown in the accompanying figure. Measurements (in feet) across the spill are made at 5-foot intervals, as indicated
The life span X of a certain electrical appliance is exponentially distributed with density functionwhere x measures time (in months). The appliance carries a 1-year warranty from the manufacturer. Neva purchases one of the these appliances, selected at random from the manufacturer’s stock. Find
One way to find the average temperature for a day is to take temperature readings at evenly spaced intervals for 24 hours and then find the average of those readings. A more sophisticated approach is to use Simpson’s rule and the formula for finding the average value of a function from Section
It is estimated that t years from now, a certain investment will be generating income at the rate of f(t) = 8,000 + 400t dollars per year. If the income is generated in perpetuity and the prevailing annual interest rate remains fixed at 5% compounded continuously, find the present value of the
Suppose the time between the arrivals of successive cars at a toll booth is measured by a random variable X that is exponentially distributed with the density functionwhere x measures time (in minutes). Find the probability that a randomly selected pair of successive cars will arrive at the tool
In Exercises 44 and 45, the population density p(r) of an urban area is given. In each case, use an improper integral to estimate the total population of the urban area.p(r) = 100e−0.02r
In Exercises 44 and 45, the population density p(r) of an urban area is given. In each case, use an improper integral to estimate the total population of the urban area.p(r) = 100e−0.02r + 2,000e−0.001r2
A certain nuclear power plant produces radioactive waste at the rate of 600 pounds per year. The waste decays exponentially at the rate of 2% per year. How much radioactive waste from the plant will be present in the long run?
The accompanying table gives the number of reported deaths due to AIDS, during the t th year after 1995, for the period 1995 to 2006. (Source: Centers for Disease Control and Prevention, National Center for HIV, STD, and TB Prevention.)However, the table does not tell the complete story because
A new strain of influenza has just been declared an epidemic by health officials. Currently, 3,000 people have the disease and new victims are being added at the rate of R(t) = 50√t people per week. Moreover, the fraction of infected people who still have the disease t weeks after contracting it
In a psychological experiment, it is found that the proportion of participants who require more than t minutes to finish a particular task is given bya. Find the proportion of participants who require more than 5 minutes to finish the task.b. What proportion requires between 10 and 15 minutes to
Find the function whose tangent line has slope x ln √x for each value of x > 0 and whose graph passes through the point (2, −3).
Find the function whose tangent line has slope (x + 1)e−x for each value of x and whose graph passes through the point (1, 5).
In Exercises 50 through 53, approximate the given integral and estimate the error with the specified number of subintervals using:(a) The trapezoidal rule(b) Simpson’s rule - dx; n = 10 X
Ann and Al are traveling in a car with a broken odometer. To determine the distance they travel between 2 and 3 P.M., Al (the passenger) takes these speedometer readings (in mph) every 5 minutes:Use the trapezoidal rule to estimate the total distance traveled by the pair during the hour in
A demographic study determines that the population density of a certain city at a distance of r miles from the city center is D(r) people per square mile (mi2), where D is as indicated in the following table for 0 ≤ r ≤ 10 at 2-mile intervals.Use the trapezoidal rule to estimate the total
A bakery turns out a fresh batch of chocolate chip cookies every 45 minutes. Hayley arrives (at random) at the bakery, hoping to buy a fresh cookie. Use an appropriate uniform density function to find the probability that Hayley arrives within 5 minutes of the time (before or after) that the
Jack, the pool owner in Example 6.2.7, decides to use Simpson’s rule to recalculate the surface area of his pool by making the following measurements across the pool in 6-ft intervals instead of the 4-ft intervals shown in Figure 6.6. What is the recalculated area?Data from Example 6.2.7Jack
A mathematical model in political science* asserts that the length of time served (continuously) by a legislator is a random variable X that is exponentially distributed with density functionwhere t is the number of years of continuous service and c is a positive constant that depends on the nature
Let X be the random variable that represents the number of years a patient with a particular type of cancer survives after receiving radiation therapy. Suppose X has an exponential distribution with parameter k. Research indicates that a patient with this type of cancer has an 80% chance of
In Exercises 50 through 53, approximate the given integral and estimate the error with the specified number of subintervals using:(a) The trapezoidal rule(b) Simpson’s rule et² dx; n = 8
After t hours on the job, a factory worker can produce 100te−0.5t units per hour. How many units does the worker produce during the first 3 hours?
A manufacturer has found that marginal cost is (0.1q + 1)e0.03q dollars per unit when q units have been produced. The total cost of producing 10 units is $200. What is the total cost of producing the first 20 units?
Let X be the random variable that represents the time (in hours) between successive visits by hummingbirds to feed on the flowers of a particular tropical plant. Suppose X is distributed exponentially with parameter λ = 0.5. Find the probability that the time between the arrival of successive
Money is transferred into an account at the rate of R(t) = 3,000 + 5t dollars per year for 10 years, where t is the number of years after 2000. If the account pays 5% interest compounded continuously, how much will be in the account at the end of the 10-year investment period (in 2010)?
A 2-hour movie runs continuously at a local theater. Priya leaves for the theater without first checking the show times. Use an appropriate uniform density function to find the probability that she will arrive at the theater within 10 minutes of the start of the film (before or after).
Show that an exponentially distributed random variable X with probability density functionhas expected value E(X) = 1/k. f(x) = = [ke-kx lo if x ≥ 0 if x < 0
After t years of operation, a certain nuclear power plant produces radioactive waste at the rate of R(t) pounds per year, where R(t) = 300 − 200e−0.03tThe waste decays exponentially at the rate of 2% per year. How much radioactive waste from the plant will be present in the long run?
An investment will generate income continuously at the rate of R(t) = 20 + 3t hundred dollars per year for 5 years. If the prevailing interest rate is 7% compounded continuously, what is the present value of the investment?
Joel Evans, the manager of a national chain of pizza parlors, is selling a 6-year franchise to operate its newest outlet in Orlando, Florida. Experience in similar localities suggests that t years from now, the franchise will be generating profit continuously at the rate of R(t) = 300 + 5t thousand
Answer the questions in Exercise 51 for the demand functionwhen 12,000 (q0 = 12) units are demanded.Data from Exercises 51A manufacturer has determined that when q thousand units of a particular commodity are produced, the price at which all the units can be sold is p = D(q) dollars per unit, where
In Exercises 50 through 53, approximate the given integral and estimate the error with the specified number of subintervals using:(a) The trapezoidal rule(b) Simpson’s rule 2 et dx; n = 10
A manufacturer has determined that when q thousand units of a particular commodity are produced, the price at which all the units can be sold is p = D(q) dollars per unit, where D is the demand functiona. At what price are 5,000 (q0 = 5) units demanded?b. Find the consumers’ surplus when 5,000
The median of a random variable X is the real number m such that P(X ≤ m) = 1/2. This term is used in Exercises 52 through 55.Find the median length of a telephone call in Example 6.4.4.Data from Example 6.4.4.Let X be a random variable that measures the duration of cell phone calls in a certain
In Exercises 50 through 53, approximate the given integral and estimate the error with the specified number of subintervals using:(a) The trapezoidal rule(b) Simpson’s rule 2 fixe 1 xel/* dx; n = 8
In Exercises 54 and 55, determine how many subintervals are required to guarantee accuracy to within 0.00005 of the actual value of the given integral using:(a) The trapezoidal rule(b) Simpson’s rule 3 S₁ √x dx
If your calculator has a numeric integration feature, use it to evaluate the integrals in Exercises 59 and 60. In each case, verify your result by applying an appropriate integration formula from Table 6.1. 0 279 x2V9 + 4x2dx +
Let f(x) be a continuous function with f(x) ≥ 0 for a ≤ x ≤ b. Let R be the region with area A that is bounded by the curve y = f(x), the x axis, and the lines x = a and x = b. Then the centroid (or center) of R is the point (x̅, y̅), whereIn Exercises 65 and 66, find the centroid of the
If your calculator has a numeric integration feature, use it to evaluate the integrals in Exercises 59 and 60. In each case, verify your result by applying an appropriate integration formula from Table 6.1. 2 + 3x J-19-x² dx
In Exercises 54 and 55, determine how many subintervals are required to guarantee accuracy to within 0.00005 of the actual value of the given integral using:(a) The trapezoidal rule(b) Simpson’s rule e J0.5 -1.1x dx
In a certain state, the Lorenz curves for the distributions of income for lawyers and engineers are y = L1(x) and y = L2(x), respectively, whereFind the Gini index for each curve. Which profession has the more equitable distribution of income? L₁(x) = 0.6x² + 0.4x and L₂(x)=x²³²e²-1
Use the numeric integration feature of yourbcalculator to computefor N = 1, 10, 50. Based on your results, do you think the improper integralconverges? If so, to what value? I(N) 1 - So √ / ²² ㅠ || -1² dx
Let f(x) be a continuous function with f(x) ≥ 0 for a ≤ x ≤ b. Let R be the region with area A that is bounded by the curve y = f(x), the x axis, and the lines x = a and x = b. Then the centroid (or center) of R is the point (x̅, y̅), whereIn Exercises 65 and 66, find the centroid of the
A manufacturer determines that when x hundred units of a particular commodity are produced, the profit generated is P(x) thousand dollars, whereWhat is the average profit over the production range 0 ≤ x ≤ 10? P(x) = 500 In(x + 1) (x + 1)²
After escaping from several angry camel owners (Exercise 73, Section 5.1), the spy learns that his enemy, Scelerat, is holed up in a chateau in the Alps. Disguised as an old duck plucker, the spy enters a nearby village to gather information. On the day he arrives, his identity is known only to the
The shaded region shown in the accompanying figure is a parking lot for a shopping mall. The dimensions are in hundreds of feet. To improve parking security, the mall manager plans to place a surveillance kiosk in the center of the lot (see the definition preceding Exercises 65 and 66).Where would
If your calculator has a numeric integration feature, use it to evaluate the integrals in Exercises 76 through 79. In each case, verify your result by applying an appropriate integration formula from Table 6.1. So √x² + 2x (x + 1)² dx
Repeat Exercise 57 for the curvesData from Exercises 57Use the graphing utility of your calculator to draw the graphs of the curves y = −x3 − 2x2 + 5x − 2 and y = x ln x on the same screen for x > 0. Use ZOOM and TRACE or some other feature of your calculator to find where the curves
Use the graphing utility of your calculator to draw the graphs of the curves y = −x3 − 2x2 + 5x − 2 and y = x ln x on the same screen for x > 0. Use ZOOM and TRACE or some other feature of your calculator to find where the curves intersect, and then compute the area of the region
It is projected that t years from now the population P(t) of a certain city will be changing at the rate ofthousand people per year. If the current population is 2 million, what will the population be 5 years from now? P'(t) = t In Vt+1
If your calculator has a numeric integration feature, use it to evaluate the integrals in Exercises 76 through 79. In each case, verify your result by applying an appropriate integration formula from Table 6.1. 3 J2 √4x² - 7 dx
A manufacturer determines that the marginal cost of producing q units of a particular commodity isa. Express the total cost of producing the first 8 units as a definite integral.b. Estimate the value of the total cost integral in part (a) using the trapezoidal rule with n = 8 subintervals. C'(q) =
A continuous random variable X has a Pareto distribution if its probability density function has the formwhere a and λ are parameters (positive real numbers). The Pareto distribution is often used to model the distribution of incomes in a society. Exercises 56 and 57 involve the Pareto
If your calculator has a numeric integration feature, use it to evaluate the integrals in Exercises 76 through 79. In each case, verify your result by applying an appropriate integration formula from Table 6.1. 5₁²x x² In √x dx
A continuous random variable X has a Pareto distribution if its probability density function has the formwhere a and λ are parameters (positive real numbers). The Pareto distribution is often used to model the distribution of incomes in a society. Exercises 56 and 57 involve the Pareto
Find the Gini index for an income distribution whose Lorenz curve is the graph of the function L(x) = xex−1 for 0 ≤ x ≤ 1.
If your calculator has a numeric integration feature, use it to evaluate the integrals in Exercises 76 through 79. In each case, verify your result by applying an appropriate integration formula from Table 6.1. 29 x²³√4 + 5x dx
Use the numeric integration feature of your calculator to computefor N = 10, 100, 1,000, 10,000. Based on your results, do you think the improper integralconverges? If so, to what value? I(N) 'N S₁ In(x + 1) X dx
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