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All Textbook Solutions for Calculus and Its Applications (11th Edition)

Differentiate 74. Differentiate 75. Differentiate 76. Differentiate y=ln(3x2+2x1)Differentiate 78. Differentiate 79. Differentiate f(x)=ln(x2+5X)Differentiate g(x)=exlnx2 Differentiate g(x)=e2xlnx Differentiate 83. Differentiate f(x)=ln(ex2)Differentiate g(x)=(lnx)4 (Hint: Use the Extended Power Rule.)Differentiate 86. Differentiate f(x)=ln(ln(8x))Differentiate f(x)=ln(ln(3x))Differentiate 89. Differentiate g(x)=ln(2x)ln(7x)91. Find the equation of the line tangent to the graph of at . 92. Find the equation of the line tangent to the graph of at . Find the equation of the line tangent to the graph of y=(lnx)2 at x=3.Find the equation of the line tangent to the graph of y=ln(4x27) at x=2.Business and Economics 95. Advertising. A model for consumer’s response to advertising is given by where is the number of units sold and a is the amount spent on advertising, in thousands of dollars. a. How many units were sold after spending $1000 on advertising? b. Find and c. Find the maximum and minimum values, if they exist. d. Find . Discuss whether it makes senses to continue to spend more and more on advertising. Business and Economics 96. Advertising. A model for consumer’s response to advertising is given by where is the number of units sold and a is the amount spent on advertising, in thousands of dollars. a. How many units were sold after spending $1000 on advertising? b. Find and c. Find the maximum and minimum values on N, if they exist. d. Find . Discuss . Does it make sense to spend more and more on advertising? Why or why not? An advertising model. Solve Example 10 if the advertising campaign costs $2000 per day.Business and Economics 98. An advertising model. Solve Example 10 if the advertising campaign costs $4000 per day. 99E Growth of a stock. The value, V(t), in dollars, of a share of Cypress Mills stock t months after it purchased is modeled by V(t)=58(1e1.1t)+20. a. Find V(1) and V(12) b. Find V(t). c. After how many months will the value of a share of the stock first reach $75? d. Find limtV(t). Discuss the value of a share over a long period of time. Is this trend typical?Business and Economics 101. Marginal Profit. The profit, in thousands of dollars, from the sale of x thousand candles can be estimated by . a. Find the marginal profit, b. Find , and explain what this number represents. c. How many thousands of candles should be sold to maximize profit? 102. Acceptance of a new medicine. The percentage P of doctors who prescribe a certain new medicine is where t is the time, in months. a. Find and b. Find c. How many months will it take for 90% of doctors to prescribe the new medicine? d. Find and discuss its meaning. Social Sciences 103. Forgetting. Students in a botany class took a final exam. They took equivalent forms of the exam at monthly intervals thereafter. After t months, the average score as a percentage was found to be . a. What was the average score when the students initially took the test? b. What was the average score after 4 months? c. What was the average score after 24 months? d. What percentage of their original answers did the students retain after 2 years (24 months)? e. Find f. Find the maximum value, if one exits. g. Find and discuss its meaning. Social Sciences 104. Forgetting. As part of a study, students in a psychology class took a final exam. They took equivalent forms of the exam at monthly intervals thereafter. After t months, the average score as a percentage, was found to be given by . a. What was the average score when they initially took the test, ? b. What was the average score after 4 months? c. What was the average score after 24 months? d. What percentage of their original answers did the students retain after 2 years (24 months) e. Find f. Find the maximum and minimum values, if they exist. g. Find , and discuss its meaning. Social Sciences Walking speed. Bornstein and Bornstein found in a study that the average walking speed v, in feet per second, of a person living v(p)=0.37lnp+0.05. a. The population of Seattle is 635,000(p=6352). What is the average walking speed of a person living in Seattle? b. The population of New York is 8,340,000. What is the average walking speed of a person living in New York? c. Find v(p). d. Interpret v(p) found in part (c).Social Sciences Hullian learning model. A keyboarder learns to type W words per minute after t weeks of practice, where W is given by W(t)=100(1e0.3t). a. Find W(1) and W(8) b. Find W(t) c. After how many weeks will the keyboarders speed be 95 words per minute? d. Find limtW(t), and discuss its meaning.107. Solve for t. Differentiate. f(x)=ln(x3+1)5 Differentiate. 109. Differentiate. 110. Differentiate. 111. Differentiate. f(x)=log5x Differentiate. f(x)=log7x Differentiate. y=ln5+x2 116E 117E 118E To prove Proprieties P1, P2, P3, and P7 of Theorem 3, let X=logaM and Y=logaN, and give reasons for the steps listed in Exercises 119 122. Proof of P1 of Theorem 3 M=aX and N=aY, definition of logarithm So MN=aXaY=aX+y Product Rule for exponents Thus, loga(MN)=X+Y definition of logarithm =logaM+logaN. substitution To prove Proprieties P1, P2, P3, and P7 of Theorem 3, let X=logaM and Y=logaN, and give reasons for the steps listed in Exercises 119 122. Proof of P2 of Theorem 3. M=aX and N=aY, definition of logarithm So MN=aXaY=aXY Quotient Rule for exponents Thus, logaMN=XY Definition of logarithm =logaM+logaN substitution To prove Proprieties P1, P2, P3, and P7 of Theorem 3, let X=logaM and Y=logaN, and give reasons for the steps listed in Exercises 119 122. Proof of P3 of Theorem 3 M=aX Definition of logarithm So, Mk=(aX)k if u=v, then uc=vc =aXk Power Rule for exponents Thus, logaMk=Xk definition of logarithm =klogaM. substitution and the commutative law for multiplication To prove Proprieties P1, P2, P3, and P7 of Theorem 3, let X=logaM and Y=logaN, and give reasons for the steps listed in Exercises 119 122. Proof of P7 of Theorem 3 Let logbM=R Then bR=M, definition of logarithm and loga(bR)=logaM. Ifu=v,thenlogcu=logcv_ Thus, Rlogab=logaM. using Property P3 and R=logaMlogab. Ifa=bandc0thena/c=b/c_ It follows that logbM=logaMlogab. substitution 124E 125E 126. Explain why is not defined. (Hint: Rewrite it as an equivalent exponential expression.) 127E 128E 129E 1. Find the general form of if . 2. Find the general form of g if. 3. Find the general form of the function that satisfies. Find the general form of the function that satisfies dP/dt=3P(t).Find the general form of the function that satisfies dQ/dt=kQ.Find the general form of the function that satisfies dR/dt=kR.U.S. patents. The number of applications for patents, N, grew dramatically in recent years, with growth averaging about 5.8% per year. That is, N(t)=0.058N(t). (Source: Based on data from the U.S. Patent and Trademark office, 20092013.) a. Find the function that satisfies the equation. Assume that t=0 corresponds to 2009, when approximately 483,000 patent applications were received. b. Estimate the number of patent applications in 2020. c. Estimate the doubling time for N(t).8. Franchise Expansion. Pete Zah’s is selling franchises for pizza shops throughout the county. The marketing manager estimates that the number of franchises, N, will increase at the rate of 10% per year, that is, a. Find the function that satisfies this equation. Assume that the number of franchises at. b. How many franchises will there be in 20 yr? c. In what period of time will the initial number of 50 franchises double? Compound Interest. If an amount P0 is invested in the Mandelbrot Bond Fund and interest is compounded continuously at 5.9% per year, the balance P grows at the rate given by dPdt=0.059P. a. Find the function that satisfies the equation. Write it in terms of P0 and 0.059. b. Suppose $1000 is invested. What is the balance after 1 yr? After 2 yr? c. When will an investment of $1000 double itself?10. Compound interest. If an amount is invested in a savings account and interest is compounded continuously at 4.3% per year, the balance P grows at the rate given by . a. Find the function that satisfies the equation. Write it in terms of and 0.043. b. Suppose $20,000 is invested. What is the balance after 1 yr? After 2 yr? c. When will an investment of $20,000 double itself? 11. Bottled Water Sales. Since 2000, sales of bottled water have increased at the rate of approximately 9.3% per year. That is, the volume of bottled water sold, G, in billions of gallons, t years after 2000 is growing at the rate given by . (Source: The Beverage Marketing Corporation.) a. Find the function that satisfies the equation, given that approximately 4.7 billion gallons of bottled water were sold in 2000. b. Predict the number of gallons of water sold in 2025. c. What is the doubling time for? Annual net sales. Green Mountain Coffee Roasters produces many varieties of flavored coffees, teas, and K-cups. Since 2008, the net sales S of the company have grown exponentially at the rate of 46.3% per year. The grown can be approximated by dSdt=0.463S, where t is the number of years since 2008. (Source: Green Mountain Coffee Roasters financial statements.) a. Find the function that satisfies the equation, given that net sales in 2008 (t=0) were approximately $500,000. b. Estimate net sales in 2012 and 2016. c. What is the doubling time for S(t)?Annual interest rate. Euler Bank advertises that it compounds interest continuously and that it will double your money in 15 yr. What is its annual interest rate?14. Annual interest rate. Hardy Bank advertises that it compounds interest continuously and that it will double your money in 12 yr. What is its annual interest rate? Oil demand. The growth rate of the demand for oil in the United States is 10% per year. When will the demand be double that of 2012?Coal demand. The growth rate of the demand for coal in the world is 4% per year. When will the demand be double that of 2006?Interest compounded continuously. For Exercises 17-20, complete the following. 17. Initial Investment at Interest Rate, k Doubling Time, T (in years) Amount after 5 yr $75,000 6.2% 11.2 $102,256.88 Interest compounded continuously. For Exercises 17-20, complete the following. Initial Investment at t=0,P0 Interest Rate, k Doubling Time, T (in years) Amount after 5 yr $5,000 7.1% 9.8 $7,130.90 Interest compounded continuously. For Exercises 17-20, complete the following. Initial Investment at t=0,P0 Interest Rate, k Doubling Time, T (in years) Amount after 5 yr $7, 500 8.4% 8.3 $11,414.71 Interest compounded continuously. For Exercises 17-20, complete the following. Initial Investment at t=0,P0 Interest Rate, k Doubling Time, T (in years) Amount after 5 yr $12, 800 6.3% 11 $17,539.32 21. Art masterpieces. In 2004, a collector paid $104,168,000 for Pablo Picasso’s “Garcon a la Pipe”. The same painting sold for $30,000 in 1950. (Source: BBC News, 5/6/04) Boy with a Pipe (1905), Pablo Picasso ©2011 Picasso Estate/ARS a. Find the exponential growth rate k, to three decimal places, and determine the exponential growth function V, for which is the painting’s value, in dollars, t years after 1950. b. Predict the value of the painting in 2015. c. What is the doubling time for the value of the painting? d. How long after 1950 will the value of the painting be $1 billion? 22. Per capita income. In 2009, U.S. per capita personal income I was $48,040. In 2012, it was $52,430. (Source: data.worldbank.org.) Assume that the growth of U.S. per capita personal income follows an exponential model. a. Letting be 2009, write the function. b. Predict what U.S. per capita income will be in 2020. c. In what year will U.S. per capita income be double that of 2009? 23. Federal receipts. In 2011, U.S. federal receipts (money taken in) totaled $2.30 trillion. In 2013, total federal receipts were $2.77 trillion. (Source: usgovernmentrevenue.com.) Assume that the growth of total federal receipts, F, can be modeled by an exponential function and use 2011 as the base year. a. Find the growth rate k to six decimal places, and write the exponential function, for total receipts in trillions of dollars. b. Estimate total federal receipts in 2015. c. When will total federal receipts be $10 trillion? Consumer price index. The consumer price index compares the costs, c, of goods and services over various years, where 1983 is used as a base (t=0). The same goods and services that cost $100 in 1983 cost $226 in 2012. a. Model c as an exponential function rounding the growth rate k to five decimal places. b. Estimate what the goods and services costing $100 in 1983 will cost in 2020. c. In what year did the same goods and services cost twice the 1983 price?Total mobile data traffic. The following graph shows the predicted monthly mobile data traffic for the years 20132018. Use these data in Exercises 25 and 26. MOBILE DATA TRAFFIC a. Use regression to fit an exponential function y=abx to the data. Let y be the monthly mobile data traffic in exabytes (EB, 1 exabyte =1 billion gigabytes) and x the number of years since 2013. Then convert the function to an exponential function, base e, using the fact that b=elnb. b. What is the exponential growth rate, as a percentage? c. Estimate the monthly mobile data traffic in 47.3% 2020. d. When will the monthly mobile data traffic exceed 50 exabytes? e. What is the doubling time for monthly mobile data traffic?Total mobile data traffic. The following graph shows the predicted monthly mobile data traffic for the years 20132018. Use these data in Exercises 25 and 26. MOBILE DATA TRAFFIC a. Find an exponential function, base e, that fits the data, using the points (0,1.5) and (2,4.4). Let x represent the number of years since 2013. b. Estimate the total monthly mobile data traffic in 2019 and 2022. c. When will total monthly mobile data traffic exceed 50 exabytes? d. What is the doubling time for total monthly traffic? e. Compare your answers with those from parts (c) (e) of Exercise 25. Decide which exponential function seems to fit the data better, and explain why.Value of Manhattan Island. Peter Minuit of the Dutch West India Company purchased Manhattan Island from the natives living there in 1626 for $24 worth of merchandise. Assuming an exponential rate of inflation of 5%, how much will Manhattan be worth in 2020?28. Total Revenue. Intel, a computer chip manufacturer, reported $1265 million in total revenue in 1986. In 2012, the total revenue was $53.3 billion. (Source: intel.com.) Assuming an exponential model, find the growth rate k, to four decimal places, and determine the revenue function R, with in billions of dollars. Then predict the company’s total revenue for 2020. 29. The U.S. Forever Stamp. The U.S. Postal Service sells the Forever Stamp, which is always valid as first-class postage on standard envelopes weighting 1 ounce or less, regardless of any subsequent increases in the first-class rate. (Source: U.S. Postal Service.) a. The cost of a first-class postage stamp was in 1962 and 49 in 2014. This increase represents exponential growth. Write the function S for the cost of a stamp t years after 1962 . b. What was the growth rate in the cost? c. Predict the cost of a first-class postage stamp in 2016, 2018, and 2020. d. An advertising firm spent $4900 on 10,000 first-class postage stamps at the beginning of 2014. Knowing it will need 10,000 first-class stamps in each of the years 2015-2024, it decides to try to save money by also buying enough stamps to cover those years at the time of the 2014 purchase. Assuming there is a postage increase in each of the years 2016, 2018, and 2020 to the cost predicted in part (c), how much money will the firm save by buying the Forever Stamps in 2014? e. Discuss the pros and cons of the purchase decision described in Part (d). 30E Effect of advertising. Suppose that SpryBorg Inc. introduces a new computer game in Houston using television advertisements. Surveys show that P% of the target audience buys the game after x ads are broadcast, satisfying P(x)=1001+49e0.13x. a. What percentage buys the game without seeing a TV ad (x=0)? b. What percentage buys the game after the ad is run 5 times? 10 times? 20 times? 50 times? 60 times? c. Find the rate of change, P(x). d. Sketch a graph of the function.Cost of a Hershey bar. The cost of a Hershey bar was $0.05 in 1962 and $0.99 in 2013 (in a supermarket, not in a movie theater). a. Find an exponential function that fits the data. b. Predict the cost of a Hershey bar in 2020 and 2025.Superman comic book. In August 2014, a 1938 comic book featuring the first appearance of Superman sold at auction for a record price of $3.2 million. The comic book originally cost 10(0.10). (Sources: eBay; money.cnn.com.) Use the two data points (0,0.10) and (76,3,200,000), and assume that the value V of the comic book has grown exponentially, as given by dVdt=kV. (In the summer of 2010, a family faced foreclosure on their mortgage. As they were packing, they came across some old comic books in the basement, and one of them was a copy of this first Superman comic. They sold it and saved their house.) a. Find the function that satisfies this equation. Assume that V0=0.10 b. Estimate the value of the comic book in 2020 c. What is the doubling time for the value of the comic book? d. After what time will the value of the comic book be $30 million, assuming there is no change in the growth rate?34. Batman comic book. Refer to Example 6. In what year will the value of the comic book be $5 million? Batman comic book. Refer to Example 6. In what year will the value of the comic book be $10 million?Population Growth For Exercise 36-40, complete the following. 36. Population Exponential Growth Rate, k Doubling Time, T(in years) Mexico 3.5%/yr 19.8 Population Growth For Exercise 36-40, complete the following. 37. Population Exponential Growth Rate, k Doubling Time, T(in years) Europe 1%/yr 69.31 Population Growth For Exercise 36-40, complete the following. Population Exponential Growth Rate, k Doubling Time, T(in years) Oil Reserves 10%/yr 6.931 Population Growth For Exercise 36-40, complete the following. Population Exponential Growth Rate, k Doubling Time, T(in years) Coal Reserves 4%/yr 17.3 Population Growth For Exercise 36-40, complete the following. 40. Population Exponential Growth Rate, k Doubling Time, T(in years) Alaska 2.794%/yr 24.8 Bicentennial growth of the United States. The population of the United States in 1776 was about 2,508,000. In the countrys bicentennial year, the population was about 216,000,000. a. Assuming an exponential model, what was the growth rate of the United States through its bicentennial year? b. Is exponential growth a reasonable assumption? Why or why not?Limited population growth: Human Population. Seventeen adults came ashore from the British ship HMS Bounty in 1970 to settle on the uninhabited South Pacific Island Pitcairn. The population, P(t), of the island t years after 1790 can be approximated by the logistic equation P(t)=340017+183e0.982t (Source: www.government.pn) a. Find the population of the Island after 10 yr, 50 yr, and 75 yr. b. Find the rate of change in the population, P(t). c. Find the rate of change in the population after 10 yr, 50 yr, and 75 yr. d. What is the limiting value for the population of Pitcairn? (The limiting value is the number to which the population gets closer and closer but never reaches.)43. Limited population growth: tortoise population. The tortoise population, in a square mile of the Mojave Desert after t years can be approximated by the logistic equation (Source: www.deserttortoise.org) a. Find the tortoise population after 0 yr, 5 yr, 15 yr, and 25 yr. b. Find the rate of change in population, . c. Find the rate of change in the population after 0 yr, 5 yr, 15 yr, and 25 yr. d. What is the limiting value (see Exercise 42) for the population of tortoises in a square mile of the Mojave Desert? 44. Limited population growth. A lake is stocked with 400 rainbow trout. The size of the lake, the availability of food, and the number of other fish restrict population growth to a limiting value of 2500 trout. (see Exercise 42). The population of trout in the lake after time t, in months, is approximated by . a. Find the population after 0 months, 1 months, 5 months, 10 months, 15 months, and 20 months. b. Find the rate of change, . c. Sketch a graph of the function. Women college graduates. The number of women earning a bachelors degree from a 4-yr college in the United States grew from 48,869 in 1930 to approximately 920,000 in 2010. (Source: National Center for Education Statistics.) Find an exponential function that fits the data, and the exponential growth rate, rounded to the nearest hundredth of a percent.Hullian learning model. The Hullian learning model asserts that the probability p of mastering a task after t learning trials is approximated by p(t)=1ekt, where k is a constant that depends on the task to be learned. Suppose a new dance is taught to an aerobics class. For this particular dance, assume k=0.28. a. What is the probability of mastering the dance in 1 trial? 2 trial? 5 trials? 11 trials? 16 trials? 20 trials? b. Find the rate of change, p(t). c. Sketch a graph of the function.Spread of infection. Spread by skin-to-skin contact or via shared towels or clothing, methicillin-resident Staphylococcus aureus (MRSA) can easily infect growing numbers at a university. Left unchecked, the number of cases of MRSA on a university campus t weeks after the first 9 cases occur can be modeled by N(t)=568.8031+62.200e0.092t. (Source: Vermont Department of Health, Epidemiology Division.) 11.8, 221.4, 547.2 a. Find the number of infected students beyond the first 9 cases after 3 weeks, 40 weeks, and 80 weeks. b. Find the rate at which the disease is spreading after 20 weeks. c. Explain why an unrestricted growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation.48. Diffusion of information. Pharmaceutical firms invest significantly in testing new medications. After a drug is approved by the Federal Drug Administration, it still takes time for physicians to fully accept and start prescribing it. The acceptance by physicians approaches a limiting value of 100%, or 1, after t months. Suppose that the percentage P of physicians prescribing a new cancer medication after t months is approximated by . 0%, 33%, 55%, 70%, 86%, 99.2%, 99.8% a. What percentage of doctors are prescribing the medication after 0 months? 1 month? 2 month? 3 months? 5 months? 12 months? 16 months? b. Find , and interpret its meaning. c. Sketch a graph of the function. 49. Spread of a rumor. The rumor “People who study math all get scholarship” spreads across a college campus. Data in the following table show the number of students N who have heard the rumor after time t, in days. Time, t (in days) Number, N, Who Heard the Rumor 1 1 2 2 3 4 4 7 5 12 6 18 7 24 8 26 9 28 10 28 11 29 12 30 a. Use REGRESSION to fit a logistic equation, to the data. b. Estimate the limiting value of the function. At most, how many students will hear the rumor? c. Graph the function. d. Find the rate of change, . e. Find and explain its meaning. We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The growth in value of a U.S. savings bond We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. 51. The growth in the length of Zachary’s hair following a haircut We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The growth in sales of electric cars We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The drop and rise of a lakes water level during and after a drought We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The rapidly growing sales of organic foods We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. 55. The number of manufacturing jobs that have left the United States since 1995 We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. The life expectancy of the average American We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. 57. The occupancy (number of apartments rented) of a newly opened apartment complex We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. 58. The decrease in population of a city after its principal industry closes We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. 59. The weight of a dog from birth to adulthood We have now studied models for linear, quadratic, exponential, and logistic growth. In the real world, understanding which is the most appropriate type of model for a given situation is an important skill. For each situation in Exercises 50-60, identify the most appropriate type of model and explain why you chose that model. List any restrictions you would place on the domain of the function. 60. The efficiency of an employee as a function of number of hours worked 61. Find an expression relating the exponential growth rate k and the quadrupling time. Find an expression relating the exponential growth rate k and the tripling time T3.63. Quantity grows exponentially with a doubling time of 1 yr. Quantity grows exponentially with a doubling time of 2 yr. If the initial amounts of and are the same, how long will it take for to be twice the size of? 64. To what exponential growth rate per hour does a growth rate of 100% per day correspond? 65. Complete the table below, which relates growth rate k and doubling time T. Growth Rate, k (per year) 1% 2% 14% Doubling Time, T (in years) 15 10 Graph. Is this a linear relationship? Explain. Describe the differences in the graphs of an exponential function and a logistic function. The Rule of 70. The relationship between doubling time T and growth rate k is the basis of the Rule of 70. Since T=ln2k=0.693147k=69.3147100k70100k, we can estimate the length of time needed for a quantity to double by dividing the growth rate k (expressed as a percentage) into 70.Estimate the time needed for an amount of money to double, if the interest rate is 7%, compounded continuously.68. Estimate the time needed for the population in a city to double, if the growth rate is 3.5%, compounded continuously. Using a calculator, find the exact doubling times for the amount of money in Exercise 67 and the population in Exercise 68.70. Describe two situations where it would be preferable to use the Rule of 70 instead of the formula. Explain why it would be acceptable to use this rule in these situations. Business: total revenue. The revenue of Red Rock, Inc., in millions of dollars, is given by the function R(t)=40001+1999e0.5t were t is measured in years. where t is measured in years. a. What is R(0), and what does it represent? b. Find limtR(t). Call this value Rmax, and explain what it means. c. Find the value of t (to the nearest integer) for which R(t)=0.99Rmax.In Exercise 1-8, find the half-line for each situation. An element loses 12% of its mass every year.In Exercise 1-8, find the half-line for each situation. An population of bacteria decreases by 5.75% per month.In Exercise 1-8, find the half-line for each situation. A vehicle loses 0.8% of its value every month.In Exercise 1-8, find the half-line for each situation. A city loses 3.9% of its population every year.In Exercise 1-8, find the half-line for each situation. 5. The value of a dollar decreases by 3% every year. . In Exercise 1-8, find the half-line for each situation. An element loses 1.75% of its mass every day..In Exercise 1-8, find the half-line for each situation. 7. An investment loses 1.9% of its value every week. In Exercise 1-8, find the half-line for each situation. 8. A motor home loses 0.5% of its value every week. Life and Physical Sciences Radioactive Decay. Iodine-131 has a decay rate of 9.6% per day. The rate of change of an amount N of iodine-131 is given by dNdt=0.096N, where t is the number of days since decay begin. a. Let N0 represent the amount of iodine-131 present at t=0. Find the exponential function that models the situation. b. Suppose 500 g of iodine-131 is present at t=0. How much will remain after 4 days? c. After how many days will half of the 500 g of iodine-131 remain?Life and Physical Sciences 10. Radioactive Decay. Carbon-14 has a decay rate of 0.012097% per year. The rate of change of an amount N of carbon-14 is given by where t is the number of years since decay began. a. Let represent the amount of carbon-14 present at . Find the exponential function that models the situation. b. Suppose 200 g of carbon-14 is present at . How much will remain after 800 yr? c. After how many years will half of the 200 g of Carbon-14 remain? Life and Physical Sciences 11. Chemistry. Substance A decomposes at a rate proportional to the amount of A present. a. Write an equation that gives the amount A left of an initial amount after time t. b. It is found that 10 lb of A will reduce to 5 lb in 3.3 hr. After how long will there be only 1 lb left? Life and Physical Sciences Chemistry. Substance A decomposes at a rate proportional to the amount of A present. a. Write an equation that gives the amount A left of an initial amount A0 after time t. b. It is found that 8 g of A will reduce to 4 g in 3hr. After how long will there be only 1 g left?Radioactive Decay. For Exercises 13-16, complete the following. 13. Radioactive Substance Decay Rate, k Half-life, T Polonium-218 23.1%/min 3 min Radioactive Decay. For Exercises 13-16, complete the following. Radioactive Substance Decay Rate, k Half-life, T Radium-226 0.0433%/yr 1600 yr Radioactive Decay. For Exercises 13-16, complete the following. 15. Radioactive Substance Decay Rate, k Half-life, T Lead-210 3.15%/yr 22 yr Radioactive Decay. For Exercises 13-16, complete the following. 16. Radioactive Substance Decay Rate, k Half-life, T Strontium-90 2.77%/yr 25 yr Half-life. Of an initial amount of 1000g of lead-210, how much will remain after 100 yr? See Exercise 15 for the value of k.Half-life. Of an initial amount of 1000g of polonium-218, how much will remain after 20 min? See Exercise 13 for the value of k.19E 20E 21. Cancer Treatment. Iodine-125 is often used to treat cancer and has a half-life of 60.1 days. In a sample, the amount of iodine-125 decreased by 25% while in storage. How long was the sample in storage? 22E Carbon Dating. Recently, while digging in Chaco Canyon, New Mexico, archaeologists found corn pollen that had lost 38.1% of its carbon-14. The age of this corn pollen was evidence that Indians had been cultivating crops in the Southwest centuries earlier than previously thought. (Source: American Anthropologist.) What was the age of the pollen?Present value. Following the birth of a child, a parent wants to make an initial investment P0 that will grow to $30,000 by the childs 20th birthday. Interest is compounded continuously at 6%. What should the initial investment be?Present value. Following the birth of their child, the Irwins want to make an initial investment P0 that will grow to $40,000 by the childs 20th birthday. Interest is compounded continuously at 5.3%. What should the initial investment be?Present value. Desmond wants to have $15,000 available in 5 yr to pay for new siding. Interest is 4.3%, compounded continuously. How much money should be invested?27. Sports salaries. An athlete signs a contract that guarantees a $9-million salary 6 yr from now. Assuming that money can be invested at 4.7%, with interest compounded continuously, what is the present value of that year’s salary? 28. Actor’s salaries. An actor signs a film contract that will pay $12 million when the film is completed 3 yr from now. Assuming that money can be invested 4.2%, with interest compounded continuously, what is the present value of that payment? 29. Estate planning. Shannon has a trust fund that will yield $80,000 in 13 yr. A CPA is preparing a financial statement for Shannon and wants to take into account the preset value of the trust fund in computing her net worth. Interest is compounded continuously at 4.8%. What is the present value of the trust fund? 30. Supply and demand. The supply and demand for stereos produced by Blaster Sound, Inc., are given by and where , is the number of stereos that the company is willing to sell for x dollars each and is the quantity that the public is willing to buy at price x. Find the equilibrium point. (See Section R.5.) Salvage value. Lucas Mining estimates that the salvage value V(t), in dollars, of a piece of machinery after t years is given by V(t)=40,000et. a. What did the machinery cost initially? b. What is the salvage value after 2 yr? c. Find the rate of change of the salvage value, and explain its meaning.32. Salvage value. Wills Investments tracks the value of a particular photocopier over a period of years. The data in the table below show the value of the copier at time t, in years, after the date of purchase. (Source International Data Corporation) Time, t (in years) Salvage Value 0 $34,000 1 22,791 2 15,277 3 10,241 4 6,865 5 4,600 6 3,084 a. Use regression to fit an exponential function to the data. Then convert that formula to where is the value when the copier is purchased and t is the time, in years, from the date of purchase. b. Estimate the salvage value of the copier after 7 yr; 10 yr. c. After what amount of time will the salvage value be $1000? d. After how long will the copier be worth half of its original value? e. Find the rate of change of the salvage value, and interpret its meaning. 33. Actuarial Science. An actuary works for an insurance company and calculates insurance premiums. Given an actual mortality rate (probability of death) for a given age, actuaries sometimes need to project future expected mortality rates of people of that age. For example, where t is the number of years into the future and is the mortality rates when projects future mortality rates. a. Suppose the actual mortality rate of a group of females aged 25 is 0.014 (14 deaths per 1000). What is the future expected mortality rate of this group of females 3, 5, and 10 yr in the future? b. Sketch the graph of the mortality function for the group in part (a) for Actuarial science. Use the formula from Exercise 33. a. Suppose the actual mortality rate of group males aged 25 is 0.023 (23 deaths per 1000). What is the future expected mortality rate of this group of males 3, 5, and 10 yr in the future? b. Sketch the graph of the mortality function Q(t) for the group in part (a) for 0t10. c. What is the ratio of the mortality rate for 25-year-old males 10 yr in the future to that for 25-year-old females 10 yr in the future (Exercise 33a)?U.S. farms. The number N of farms in the United States has declined continually since 1950. In 1950, there were 5,650,000 farms, and in 2012, that number had decreased to 2,170,000. (Sources: U.S. Department of Agriculture; National Agriculture; National Agricultural Statistics Service.) Assuming the number of farms decreased according to the exponential decay model: a. Find the value of k, and write an exponential function that describes the number of farms after time t, where t is the number of years since 1950. b. Estimate the number of farms in 2016 and in 2020. c. At this decay rate, when will only 1,000,000 farms remain?36E 37. Decline in beef consumption. Annual consumption of beef per person was about 64.6 lb in 2000 and about 61.2 lb in 2008. Assuming that the annual beef consumption t years after 2000, is decreasing according to the exponential decay model: a. Find the value of k, and write the equation. b. Estimate the consumption of beef in 2015. c. In what year (theoretically) will the consumption of beef be 20 lb per person? Population decrease of russia. The population of Russia dropped from 150 million in 1995 to 142.5 million in 2013. (Source: CIAThe World Factbook.) Assume that P(t), the population, in millions, t years after 1995, is decreasing according to the exponential decay model. a. Find the value of k, and write the equation. b. Estimate the population of Russia in 2018. c. When will the population of Russia be 100 million?Population decrease of Ukraine. The population of Ukraine dropped from 51.9 million in 1995 to 44.5 million in 2013. (Source: CIAThe World Factbook.) Assume that P(t), the population, in millions, t years after 1995, is decreasing according to the exponential decay model. a. Find the value of k, and write the equation. b. Estimate the population of Ukraine in 2018. c. In what year will the population of Ukraine be 40 million, according to this model?40. Cooling. After warming the water in a hot tub to 100°, the heating element fails. The surroundings air temperature is 40°, and in 5 min the water temperature drops to 95°. a. Find the value of the constant a in Newton’s Law of Cooling. b. Find the value of the constant k. Round to five decimal places. c. What is the water temperature after 10 min? d. How long does it take the water to cool to 41°? e. Find the rate of change of the water temperature, and interpret its meaning. 41. Cooling. The temperature in a whirlpool bath is 102°, and the room temperature is 75°. The water cools to 90° in 10 min. a. Find the value of the constant a in Newton’s Law of Cooling. b. Find the value of the constant k. Round to five decimal places. c. What is the water temperature after 20 min? d. How long does it take the water to cool to 80°? e. Find the rate of change of the water temperature, and interpret its meaning. Forensics. A coroner arrives at a murder scene at 2 a.m. He takes the temperature of the body and finds it to be 61.6. He waits 1 hr, takes the temperature again, and finds it to be 57.2. The body is in a freezer, where the temperature is 10. When was the murder committed?43. Forensics. A coroner arrives at 11 p.m. She finds the temperature of the body to be 85.9°. She waits 1 hr, takes the temperature again, and finds it to be 83.4°. She notes that the room temperature is 60°. When was the murder committed? Prisoner-of-war protest. The initial weight of a prisoner of war is 140 lb. To protest the conditions of her imprisonment, she begins a fast. Her weight t days after her last meal is approximated by W=140e0.009t. a. How much does the prisoner weight after 25 days? b. At what rate is the prisoners weight changing after 25 days?45. Political Protest. A monk weighing 170 lb begins a fast to protest a war. His weight after t days is given by . a. When the war ends 20 days later, how much does the monk weigh? b. At what rate is the monk losing weight after 20 days (before any food is consumed)? 46. Atmospheric Pressure. Atmospheric pressure P at altitude a is given by where is the pressure at sea level. Assume that (pounds per square inch). a. Find the pressure at an altitude of 1000 ft. b. Find the pressure at an altitude of 20,000 ft. c. At what altitude is the pressure? d. Find the rate of change of the pressure, and interpret its meaning. 47. Satellite power. The power supply of a satellite is a radioisotope (radioactive substance). The power output P, in watts (W), decreases at a rate proportional to the amount present and P is given by where t is the time, in days. a. How much power will be available after 375 days? b. What is the half-life of the power supply? c. The satellite cannot operate on less than 10 W of power. How long can the satellite stay in operation? d. How much power did the satellite have to begin with? e. Find the rate of change of the power output, and interpret its meaning. Cases of tuberculosis. The number of cases N of tuberculosis in the United States has decreased continually since 1956. In 1956, there were 69,895 cases. By 2012, this number had decreased by over 80%, to 10,521 cases. a. Find the value of k, and write an exponential function that describes the number of tuberculosis cases after time t, where t is the number of years since 1956. b. Estimate the number of cases in 2016 and in 2020. c. At this decay rate, in what year will there be 5000 cases?For each of the scatterplots in Exercise 49-58, determine which, if any, of these functions might be used as a model for the data: a. Quadratic: b. Polynomial, not quadratic c. Exponential: d. Exponential: e. Logarithmic: f. Logistic: 49. 50E 51E 52E 53E For each of the scatterplots in Exercise 49-58, determine which, if any, of these functions might be used as a model for the data: a. Quadratic: f(x)=ax2+bx+c b. Polynomial, not quadratic c. Exponential: f(x)=aekx,k0 d. Exponential: f(x)=aekx,k0 e. Logarithmic: f(x)=a+blnx f. Logistic: f(x)=a1+bekx 55E 56E For each of the scatterplots in Exercise 49-58, determine which, if any, of these functions might be used as a model for the data: a. Quadratic: f(x)=ax2+bx+c b. Polynomial, not quadratic c. Exponential: f(x)=aekx,k0 d. Exponential: f(x)=aekx,k0 e. Logarithmic: f(x)=a+blnx f. Logistic: f(x)=a1+bekx 58E A sample of an element lost 25% of its mass in 5 weeks. a. Find the half-life. b. After how many weeks will the sample have lost 75% of its mass?60. A vehicle lost 15% of its value in 2 yr. a. Find the half-life. b. After how many years will the vehicle be worth 40% of its original value? 61. Economics: supply and demand elasticity. The demand, and supply functions for a multipurpose printer are as follows: and . a. Find the equilibrium point. Assume that x is the price in dollars. b. Find the elasticity of demand when . The Beer-Lambert Law. A beam of light enters a medium such as water or smoky air with initial intensity. Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity I at a depth (or concentration) of x units is given by. The constant (“mu”), called the coefficient of absorption, varies with the medium. Use this law for Exercises 62 and 63. 62. Light through smog. Concentrations of particulates in the air due to pollution reduce sunlight. In a smoggy area, and x is the concentration of particulates measured in micrograms per cubic meter. What change is more significant – dropping pollution levels from to or dropping them from 60 to 50? Why? The Beer-Lambert Law. A beam of light enters a medium such as water or smoky air with initial intensity. Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity I at a depth (or concentration) of x units is given by. The constant (“mu”), called the coefficient of absorption, varies with the medium. Use this law for Exercises 62 and 63. 63. Light through sea water. Sea water has and x is measured in meters. What would increase cloudiness more – dropping x from 2m to 5m or dropping x from 7m to 10m? Explain. An interest rate decreases from 8% to 7.2%. Explain why this increases the present value of an amount due 10 yr later.66E Differentiate. 1. Differentiate. y=7x Differentiate. f(x)=8x Differentiate. 4. Differentiate. g(x)=x5(3.7)x Differentiate. g(x)=x3(5.4)x Differentiate. y=7x4+2 Differentiate. 8. Differentiate. 9. 10E Differentiate. f(x)=3x4+1 Differentiate. f(x)=127x4 Differentiate. y=log8x Differentiate. y=log4x Differentiate. y=log17x 16E Differentiate. g(x)=log32(9x2)Differentiate. g(x)=log6(5x+1)Differentiate. F(x)=log(6x7)Differentiate. 20. Differentiate. 21. Differentiate. 22. Differentiate. f(x)=4log7(x2)Differentiate. g(x)=log6(x3+5)Differentiate. 25. Differentiate. 26. Differentiate. G(x)=(log12x)5 28E Differentiate. 29. Differentiate. 30. Differentiate. y=52x31log(6x+5)32E Differentiate. 33. Differentiate. 34. Differentiate. f(x)=(3x5+x)5log3x Differentiate. g(x)=x3x(log5x)Double declining balance depreciation. An office machine is purchased for $5200. Under certain assumptions, its salvage value, V, in dollars, is depreciated according to a method called double declining balance, by basically 80% each year, and is given by V(t)=5200(0.80)t, where t is the time, in years, after purchase. a. Find V(t). b. Interpret the meaning of V(t).Recycling aluminum cans. It is known that 45% of all aluminum cans distributed are recycled each year. A beverage company uses 250,000 lb of aluminum cans. After recycling, the amount of aluminum, in pounds, still in use after t years is given by N(t)=250,000(0.45)t. (Source: The Container Recycling institute.) a. Find N(t). b. Interpret the meaning of N(t).39. Recycling glass. In 2012, 34.1% of all glass containers were recycled. A beverage company used 400,000 lb of glass containers per year. After recycling, the amount of glass, in pounds, still in use after t years is given by . (Source: www.gpi.org.) a. Find. b. Interpret the meaning of. Household liability. The total financial liability, in billions of dollars, of U.S. households can be modeled by L(t)=1547(1.083)t, where t is the number of years after 1980. The graph of this function follows. a. Using this model, predict the total financial liability of U.S. households in 2012. b. Find L(25). c. Interpret the meaning L(25).Small Business. The number of nonfarm proprietorships, in thousands, in the United States can be modeled by N(t)=8400lnt10,500, where t is the number of years after 1970. The graph of this function is given below. a. Using this model, predict the number of nonfarm proprietorships in the United States in 2020. b. Find N(45). c. Interpret the meaning of N(45).Annuities. Yukiko opens a savings account to pay for her new babys college education. She deposits $200 every month into the account at an annual interest rate of 4.2%, compounded monthly. $22,772.48 a. Find A(t), the value of Yukikos account after t years. b. What is the value of Yukikos account after 8 yr? c. What is the rate of change in the value of Yukikos account after 8 yr?43. Annuities. Nasim opens a retirement savings account, and she deposits $1000 every 4 months into the account, which has an annual interest rate of 3.8%, compounded every 4 months. $45,255.66 a. Find the value of Nasim’s account after t years. b. What is the value of Nasim’s account after 12 yr? c. What is the rate of change in the value of Nasim’s account after 12 yr? 44E The magnitude R (measured on the Richter scale) of an earthquake of intensity I is defined as R=logII0, where I0 is a minimum intensity used for comparison. If one earthquake is 10 times as intense as another, its magnitude on the Richter scale is 1 higher; if one eqrthquake is 100 times as intense as another, its magnitude is 2 higher, and so on. Thus, an earthquake whose magnitude is 6 on the Richter scale is 10 times as intense as an earthquake whose magnitude is 5 and 100 times as intense as an earthquake whose magnitude is 4. Use this information for Exercises 45 and 46. On January 12, 2010, a devastating earthquake struck the Caribbean nation of Haiti. It had an intensity that was 10,000,000, or 107, times as intense as I0. What was this earthquakes magnitude on the Richter scale? (Hint: Let I=107I0)The magnitude R (measured on the Richter scale) of an earthquake of intensity I is defined as where is a minimum intensity used for comparison. If one earthquake is 10 times as intense as another, its magnitude on the Richter scale is 1 higher; if one earthquake is 100 times as intense as another, its magnitude is 2 higher, and so on. Thus, an earthquake whose magnitude is 6 on the Richter scale is 10 times as intense as an earthquake whose magnitude is 5 and 100 times as intense as an earthquake whose magnitude is 4. Use this information for Exercises 45 and 46. 46. Find the magnitude of an earthquake that is 5,600,000 times as intense as. If two earthquakes have magnitudes R1 and R2, where R1R2, their relative intensity is given by R1R2=logI1I2. Thus, comparing an earthquake of magnitude 8.0 with another earthquake of magnitude 5.5, we have 8.05.5=logI1I2 or 2.5=logI1I2, and I1=102.5I2. Since 102.5316, an earthquake of magnitude 8.0 is about 316 times as intense as an earthquake of magnitude 5.5. Use this information for Exercises 47 and 48. The following table shows the magnitudes of selected large earthquakes. Earthquake Magnitude Sumatran-Andaman, 2004 9.2 Japan, 2011 9.0 San Francisco, 1906 8.0 Baja California, 2010 7.2 San Fernando, 1971 6.6 a. How many times more intense was the Japanese earthquake of 2011 than the Baja California earthquake of 2010? b. How many times more intense was the Sumatran-Andaman earthquake of 2004 than the San Fernando earthquake of 1971?48E 49E 50E 51E 52E 53E Finding Natural Logarithms as Limits. Given that the derivative of f(x)=ax is f(x)=ax(lna), in Section 3.1, we showed that f(x)=axlimh0ah1h. Thus, we can define lna=limh0ah1h. In Exercises 54-57, use this definition to find each limit. limh03h1h Finding Natural Logarithms as Limits. Given that the derivative of f(x)=ax is f(x)=ax(lna), in Section 3.1, we showed that f(x)=axlimh0ah1h. Thus, we can define lna=limh0ah1h. In Exercises 54-57, use this definition to find each limit. limh05h1h Finding Natural Logarithms as Limits. Given that the derivative of f(x)=ax is f(x)=ax(lna), in Section 3.1, we showed that f(x)=axlimh0ah1h. Thus, we can define lna=limh0ah1h. In Exercises 54-57, use this definition to find each limit. limh0eh1h Finding Natural Logarithms as Limits. Given that the derivative of is , in Section 3.1, we showed that . Thus, we can define . In Exercises 54-57, use this definition to find each limit. 57. Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given in Exercises 58-65. f(x)=3(2x)Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given in Exercises 58-65. 59. 60E 61E 62E 63E Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given in Exercises 58-65. y=logaf(x), for f(x) positive Use the Chain Rule, implicit differentiation, and other techniques to differentiate each function given in Exercises 58-65. 65. , for positive 66. Consider the function, with. a. Find. (Hint: Take the natural logarithm of both sides and differentiate implicitly.) b. Find the minimum value of y on. 67E 68E In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 1. , compounded monthly In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. P=8000;r=5;t=8yr, compounded monthly In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. P=12,000;r=5.7;t=6yr, compounded quarterly In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 4. , compounded quarterly In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 5. , compounded monthly In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. P=800;r=3.8;t=2yr, compounded monthly In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 7. , compounded semiannually In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 8. , compounded semiannually In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 9. , compounded annually In Exercises 1-10, find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. 10. , compounded annually Car loans. Todd purchase a new Honda Accord LX for $22,150. He makes a $4000 down payment and finances the reminder through an amortized loan at an annual interest rate of 6.5%, compounded monthly for 5 yr. a. Find Todds monthly car payment. b. Assume that Todd makes every payment for the life of the loan. Find his total payments. c. How much interest does Todd pay over the life of the loan?Car loans. Katie purchases a new Jeep Wrangler Sport for $23,000. She makes a $5000 down payment and finances the remainder through an amortized loan at an annual interest rate of 5.7% compounded monthly for 7 yr. a. Find Katies monthly car payment. b. Assume that Katie makes every payment for the life of the loan. Find her total payments. c. How much interest does Katie pay over the life of the loan?13. Home mortgages. The Hogansons purchase a new home for $195,000. They make a 25% down payment and finance the remainder with a 30-yr mortgage at an annual interest rate of 5.2%, compounded monthly. a. Find the Hogansons’ monthly mortgage payment. b. Assume that the Hogansons make every payment for the life of the loan. Find their total payments c. How much interest do the Hogansons pay over the life of the loan? Mortgages. Andre purchases an office building for $450,000. He makes a 30% down payment and finances the remainder through a 15-yr mortgage at an annual interest rate of 4.15%, compounded monthly. a. Find Andres monthly mortgage payment. b. Assume that Andre makes every payment for the life of the loan. Find his total payments. c. How much interest does Andre pay over the life of the loan?15. Credit cards. Joanna uses her credit card to finance a $500 purchase. Her card charges an annual interest rate of 22.75%, compounded monthly, and assumes a 10-yr term. Assume that Joanna makes no further purchases on her credit card. a. Find Joanna’s monthly credit card payment. b. Assume that Joanna makes every payment for the life of the loan. Find her total payments. c. How much interest does Joanna pay over the life of the loan? 16. Credit cards. Isaac uses his credit card to finance a $1200 purchase. His card charges an annual interest rate of 19.25%, compounded monthly, and assumes a 10-yr term. Assume that Issac makes no further purchases on his card. a. Find Isaac’s monthly credit card payment. b. Assume that Isaac makes every payment for the life of the loan. Find his total payments. c. How much interest does Isaac pay over the life of the loan? In Exercises 17-22, complete the first two lines of an amortization schedule for each situation, using a table as shown below. Balance Payment Portion of Payment Applied to Interest Portion of Payment Applied of Principal New Balance 17. Todd’s car loan in Exercise 11. 18E In Exercises 17-22, complete the first two lines of an amortization schedule for each situation, using a table as shown below. Balance Payment Portion of Payment Applied to Interest Portion of Payment Applied of Principal New Balance 19. The Hogansons’ home loan in Exercise 13. In Exercises 17-22, complete the first two lines of an amortization schedule for each situation, using a table as shown below. Balance Payment Portion of Payment Applied to Interest Portion of Payment Applied of Principal New Balance 20. Andre’s office building loan in Exercise 14. In Exercises 17-22, complete the first two lines of an amortization schedule for each situation, using a table as shown below. Balance Payment Portion of Payment Applied to Interest Portion of Payment Applied of Principal New Balance Joannas credit card loan in Exercise 15.In Exercises 17-22, complete the first two lines of an amortization schedule for each situation, using a table as shown below. Balance Payment Portion of Payment Applied to Interest Portion of Payment Applied of Principal New Balance Isaacs credit card loan in Exercise 16.23E Maximum loan amount. Curtis plans to purchase a new car. He qualifies for a loan at an annual interest rate of 7%, compounded monthly for 5 yr. He is willing to pay up to $200 per month. What is the largest loan he can afford?25. Maximum loan amount. The Daleys plan to purchase a new home. They qualifies for a mortgage at an annual interest rate of 4.15%, compounded monthly for 30 yr. They are willing to pay up to $1800 per month. What is the largest loan they can afford? 26E 27E 28. Comparing loan options. The Aubrys plan to finance a new home through an amortized loan of $275,000. The lender offers two options: (1) a 30-yr term at an annual interest rate of 4%, compounded monthly, and (2) a 20-yr term at an annual interest rate of 5%, compounded monthly. a. Find the monthly payments for options 1 and 2. b. Assume that the Aubrys make every monthly payment. Find their total payments for option 1 and 2. c. Assume that the Aubrys intend to make every monthly payment. Which option will result in less interest paid, and by how much? 29. Comparing Rates. Darnell plans to finance $200,000 for a new home through a 30-yr mortgage. a. Find his monthly payment, assuming an annual interest rate of 5%, compounded monthly. b. Find his monthly payment, assuming an annual interest rate of 6%, compounded monthly. c. Assume that Darnell makes every payment. How much will he save in interest over 30 yr if he selects the 5% rate? 30E 31E Retirement Planning. Kenna is 30 years old. She plans to retire at age 55 and by then wants to have saved a sum of money that will allow her to withdraw $700 per month for 25 more years (until age 80), at which time the sum will be depleted (amortized). Assume a fixed annual interest rate of 5%, compounded monthly for the entire duration. a. Find the amount of money Kenna will need at age 55. b. Find the monthly deposit Kenna needs to make in order to reach this sum, assuming that she starts saving immediately.33E 34. Structured settlement. Suppose you won a $700,000 (after taxes) settlement in court, to be paid in quarterly payments at an annual interest rate of 4%, compounded quarterly for 5 yr. Find your quarterly payment. Amortization gives the borrower an advantage: by paying more than the minimum required payment, the borrower can pay off (amortize) the principal faster and save money in interest. Assume P is the loan amount or principal, r is the interest rate, n is the compounding frequency, and t is the time in years. The amortization formula is used to determine the payment amount p that will amortize the loan in t years. However, if the borrower consistently pays Q, where Qp, the formula for the time needed to amortize the loan amount P is t=ln(nQ)ln(nQPr)nln(1+rn). Business: Home Loan. The Begays finance $200,000 for a 30-yr home mortgage at an annual interest rate of 5%, compounded monthly. a. Find the monthly payment needed to amortize this loan in 30 yr. b. Assuming that the Begays make the payment found in part (a) every month for 30 yr, find the total interest they will pay. c. Suppose the Begays pay an extra 15% every month (thus, Q=1.15.p ). Find the time needed to amortize the $200,000 loan. d. About how much total interest will the Begays pay if they pay Q every month? e. About how much will the Begays save on interest if they pay Q rather than p every month?36E 37E a. 3944. Use a spreadsheet to complete the first twelve lines of an amortization schedule for each loan in Exercises 1116. Car loans. Todd purchase a new Honda Accord LX for $22,150. He makes a $4000 down payment and finances the reminder through an amortized loan at an annual interest rate of 6.5%, compounded monthly for 5 yr. a. Find Todds monthly car payment. b. Assume that Todd makes every payment for the life of the loan. Find his total payments. c. How much interest does Todd pay over the life of the loan?a. 3944. Use a spreadsheet to complete the first twelve lines of an amortization schedule for each loan in Exercises 1116. Car loans. Katie purchases a new Jeep Wrangler Sport for $23,000. She makes a $5000 down payment and finances the remainder through an amortized loan at an annual interest rate of 5.7% compounded monthly for 7 yr. a. Find Katies monthly car payment. b. Assume that Katie makes every payment for the life of the loan. Find her total payments. c. How much interest does Katie pay over the life of the loan?41E 42E a. 39–44. Use a spreadsheet to complete the first twelve lines of an amortization schedule for each loan in Exercises 11–16. 43. Credit cards. Joanna uses her credit card to finance a $500 purchase. Her card charges an annual interest rate of 22.75%, compounded monthly, and assumes a 10-yr term. Assume that Joanna makes no further purchases on her credit card. a. Find Joanna’s monthly credit card payment. b. Assume that Joanna makes every payment for the life of the loan. Find her total payments. c. How much interest does Joanna pay over the life of the loan?

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