Word Problems with Exponents and logs



Interest Rate Problems

1. Find each of the following:

a. $500 invested at 4% compounded annually for 10 years.

b. $600 invested at 5% compounded annually for 6 years.

c. $750 invested at 3% compounded annually for 8 years.

d. $1000 invested at 4% compounded annually for 7 years.

e. $1250 invested at 3% compounded annually for 15 years.

f. $1500 invested at 4% compounded semiannually for 7 years.

g. $900 invested at 6% compounded semiannually for 5 years.

h. $950 invested at 4% compounded semiannually for 12 years.

i. $1100 invested at 5% compounded semiannually for 8 years.

j. $1750 invested at 3% compounded semiannually for 10 years.

k. $2000 invested at 5% compounded quarterly for 6 years.

l. $2250 invested at 4% compounded quarterly for 9 years.

m. $3500 invested at 6% compounded quarterly for 12 years.

n. $2500 invested at 4% compounded quarterly for 10 years.

o. $2100 invested at 5% compounded quarterly for 8 years.

p. All of the above compounded continuously.

2. What principal will amount to $2000 if invested at 4% interest compounded semiannually for 5 years?

3. What principal will amount to $3500 if invested at 4% interest compounded quarterly for 5 years?

4. What principal will amount to $3000 if invested at 3% interest compounded semiannually for 10 years?

5. What principal will amount to $2500 if invested at 5% interest compounded semiannually for 7.5 years?

6. What principal will amount to $1750 if invested at 3% interest compounded quarterly for 5 years?

7. In how many years will $300 amount to $500 at 4% compounded annually?

8. In how many years will $350 amount to $1000 at 6% compounded semiannually?

9. In how many years will $400 earn $200 in compound interest if it is invested at 5% compounded annually.

10. In how many years will $100 double itself at 4% compounded annually.

11. In how many years will $100 double itself at 4% compounded quarterly?

12. In how many years will P dollars double itself at 3% compounded semiannually?

13. A thousand dollars is left in a bank savings account drawing 7% interest, compounded quarterly for 10 years. What is the balance at the end of that time?

14. A thousand dollars is left in a credit union drawing 7% compounded monthly. What is the balance at the end of 10 years?

15. $1750 is invested in an account earning 13.5% interest compounded monthly for a 2 year period.

16. You lend out $5500 at 10% compounded monthly. If the debt is repaid in 18 months, what is the total owed at the time of repayment?

17. A $10,000 Treasury Bill earned 16% compounded monthly. If the bill matured in 2 years, what was it worth at maturity?

18. You borrow $25,000 at 12.25% interest compounded monthly. If you are unable to make any payments the first year, how much do you owe, excluding penalties?

19. A savings institution advertises 7% annual interest, compounded daily. How much more interest would you earn over the bank savings account or credit union in problems 13 and 14?

20. An 8.5% account earns continuous interest. If $2500 is deposited for 5 years, what is the total accumulated?

21. You lend $100 at 10% continuous interest. If you are repaid 2 months later, what is owed?

22. If you had a million dollars for just 2 months, how much interest could be earned in an account earning 10% compounded monthly?

23. If $1000 is invested at 16% compounded quarterly, how long will it take to quadruple?

24. If $1000 is invested at 16% compounded continuously, how long will it take to quadruple?

25. If $3600 is invested at 15% interest compounded daily, how much money will there be in 7 years? (Use a 365 day year; this is called exact interest.)

26. If $10,000 is invested at 14% interest compounded daily, how much money will there be in 6 months? (Use a 360 day year; this is called ordinary interest.)

27. A bank offers an interest rate of 7% per annum, compounded daily. What is the effective (simple) interest rate?

28. What principle should you deposit at 5.5% interest per annum, compounded semiannually, so as to have $6000 after 10 years?

29. You have two savings accounts. On account starts with a balance of $1000 and grows at 7% compounded quarterly. The other account starts at $500 and grows at 7% compounded continuously. After how many years will the account balances be equal?

30. You have $10,000 to invest over a period of 5 years. You have a choice of two investments. The first investment pays 6% per annum, compounded quarterly. The second pays 5% per annum, compounded continuously. Which is the better investment under these conditions.

31. Assuming that the Indians really did sell Manhattan for the legendary $24, and further assuming that the sale took place exactly 300 years ago, compute the amount of money the Indians would now have if they had placed the money in a bank that paid 5% interest per year, compounded quarterly, and kept it there until today.

32. Suppose that Rip van Winkle fell asleep having just deposited his $175 pay check in a savings bank offering 5% interest, compounded quarterly. How much money would he have when he woke up 20 years later?

Exponential Growth and Decay Problems

1. The half-life of U234 is 2.52 105 years. How much of a 100 gram sample remains after 10,000 years?

2. How much of a 100 gram specimen of Na22 remains after 7 years if its half-life is 2.6 years?

3. Cm242 has a half-life of 163 days. How much remains of 10 grams after one week?

4. Np239 has a half-life of 2.237 days. How much remains of 10 grams after one week?

5. How much of 10 grams of Pb189 remains after a day if its half-life is 4.98 hours?

6. How much of 25 grams of Pu234 remains after a day if its half-life is 4.98 hours?

7. What is the half-life of cesium 137 (in years) if the decay constant is

8. What is the half-life of strontium 90 (in years) if the decay constant is

9. What is the half-life of krypton (in years) if the decay constant is k = –.0641?

10. If the population of Anchorage, Alaska, continued to grow it its 1970 - 1980 rate, the city would double in size approximately every 5.4 years. Estimate its 1990 population if it was 48,081 in 1970.

11. Aurora, Colorado, would double in size every 8 years if the population continued to grow it its 1970 - 1975 rate. Estimate its 1985 population if the population was 74,974 in 1970.

12. Every 36 years, Little Rock, Arkansas would double in population if the population continued to grow at its 1960-1980 rate. Estimate the 1985 population of Little Rock if it was 107,813 in 1960.

13. Springfield, Missouri, had a population of 95,865 in 1960, and grew from 1960 to 1980 at a rate that would cause it to double every 42.23 years. Estimate Springfield's population in 1990.

14. The population of the state of Texas grew from 1950 to 1980 at an annual rate of approximately 2%. If the population in 1950 was 7,711,194, what was the population in 1980?

15. Estimate the population of Texas in 1990, using the information in problem 14.

16. Florida grew in population between 1940 and 1980 at an annual rate of 4.09%. If the population was 1,897,414 in 1940, what was the population in 1980?

17. What is the anticipated population of Florida in the year 2000 if the data in problem 16 remains constant?

18. The population of Los Angeles was 1,970,358 in 1950. It has grown since at an annual rate of 1.36%. Estimate its population in the years 1980, 1990, and 2000.

19. San Jose, CA, has had a phenomenal 6% annual growth since 1950. Estimate its population in the years 1980, 1990, and 2000, if its population was 95,280 in 1950.

20. The decay constant of Strontium-90 is –.0248. What amount of 250 mg of strontium-90 is present after 5 years?

21. Radium has a decay constant of –.0004. How much of 1000 mg of radium remains after a century?

22. The growth rate of a certain cell culture is proportional to its size. Initially, 2 × 105 cells were present. In 10 hours there were approximately 8 × 105 cells. How long will it take until there are 106 cells present.

23. The decay constant for cobalt 60 is k = –.13 when time is measured in years. Find the half-life of cobalt 60.

24. Radioactive potassium is also used for dating fossils. It has a half-life of 1.3 billion years. Determine the decay constant.

25. The size of a certain insect population is given by

P = 300e.01t

where t is measured in days. After how many days will the population equal 600?, 1200?

26. The half-life of carbon 14 is approximately 5590 years. Find the decay constant of carbon 14.

27. Some bone artifacts were found at the Lindenmeier site in Northeastern Colorado and tested for their carbon 14 content. If 25% of the original carbon 14 was still present, what is the probable age of the artifacts?

28. An artifact was discovered at the Debert site in Nova Scotia. Tests showed that 28% of the original carbon 14 was still present. What is the probable age of the artifact?

29. An artifact was found and tested for its carbon 14 content. If 12% of the original carbon 14 was still present, what is the probable age?

30. An artifact was found and tested for its carbon 14 content. If 85% of the original carbon 14 was still present, what is the probable age?

31. Sandals woven from strands of tree bark were found in Fort Rock Cave in Oregon. The bark has a carbon 14 ratio of .34 times the ratio found in living bark. Estimate the age of the sandals.

32. A 4500 year old wooden chest was found in the tomb of the twenty-fifth century B.C. Chaldean king Meskalamdug of Ur. What carbon 14 ratio would you expect to find in the wooden chest?

33. Prehistoric cave paintings were discovered in the Lascaux cave in France. Charcoal from the site was found to have a carbon 14 ratio of 15%. Estimate the age of the paintings.

34. Before radiocarbon dating was used, historians estimated that the age of the tomb of Vizier Hemaka, in Egypt, was constructed about 4900 years ago. After radiocarbon dating became available, wood samples from he tomb were analyzed and it was determined that the carbon 14 ratio was about 51%. Estimate the age of the tomb on this basis.

35. Analyses of the oldest campsites of ancient man in the Western Hemisphere reveal a carbon 14 ratio of 22.6%. Determine the probable age of the campsites.

36. The Dead Sea Scrolls are a collection of ancient manuscripts discovered in caves along the west bank of the Dead Sea. (The discovery occurred by accident when an Arab herdsman of the Taamireh tribe was searching for a stray goat.) When the linen wrappings on the scrolls were analyzed, the carbon 14 ratio was found to be 72.3%. Estimate the age of the scrolls using this information.

37. An island in the Pacific Ocean is contaminated by fallout from a nuclear explosion. If the strontium 90 is 100 times the level that scientists believe is "safe," how many years will it take for the island to once again be "safe" for human habitation? The half-life of strontium 90 is 28 years.

38. If a bacteria culture doubles in size every 20 minutes, how long will it take for a population of 104 to grow to 108 bacteria?

39. A certain cell culture grows at a rate proportional to the size of the culture. During a 10 hour experiment the culture doubled in size every three hours. At the end of the experiment approximately 105 cells were present. How many cells were present at the beginning of the experiment?

40. By 1974 the United States had an estimated 80 million gallons of radioactive products form nuclear power plants and other nuclear reactors. These waste products were stored in various sorts of containers (made of such materials as stainless steel and cement), and the containers were buried in the ground and the ocean. Scientists feel that the waste products must be prevented from contaminating the rest of the earth until more than 99.99% of the radioactivity is gone (that is, until the level is less than .0001 times the original level). If a storage cylinder contains waste products whose half-life is 1500 years, how many years must the container survive without leaking? (Note: Some of the containers are already leaking.)

41. The police were baffled by what seemed to be the perfect murder of a girl who had been found, apparently suffocated, in her kitchen. Finally, Sherlock Holmes was called in. With the aid of Dr. Watson's knowledge of botany, the mystery was solved and the following story told:

The girl had been making bread in her kitchen, whose dimensions were 10 feet by 50 feet by 10 feet. She had formed the dough into a ball of volume cubic feet and turned away to wash some dishes. At that moment Holmes' enemy, Professor Moriarty, had added a particularly virulent strain of yeast to the dough. As a result, the bread immediately started to rise, tripling in volume every 4 minutes. Before long, the dough filled the room, stopping the clock at 3:48 and squashing the girl to death against the wall. By the time Inspector Lestrade of Scotland Yard reached the scene the next day, the yeast had worked itself out and the dough returned to its original size.

At what time did Professor Moriarty add the yeast?

42. In a strange country, on the farthest moon of the nearest planet of the farthest star, is a strange race of people. In this country there is no war, disease, pestilence, famine, or inflation. And the people love each other very much. So much, in fact, that the population triples every 4 years. If the population today is 100, after how many years will the population be 106.

43. Imagine another land where the population today is 100,000 and the population triples every 5 years. When will the population in the two countries in this and the previous problem be the same?

44. After sitting unattended all winter, the Idaville municipal swimming pool is about to be reopened. Unfortunately, the town fathers discover that the water in the pool contains an unacceptable 107 bacteria per gallon. If the pool's filter can process an entire pool full of water every half-hour, and if that filter removes 75% of the bacteria in the water that passes through it, how long must the town fathers run it before the pool water reaches an acceptable level of 105 bacteria per gallon?

45. At birth the blubber beast weighs 100 lbs. Its weight grows exponentially and after 3 hours it weighs 456 lbs. When it grows to 100 times its birth weight it dies of a heart attack. How long will the average blubber beast live?

46. A particularly prolific microorganism has baffled all of modern science by dividing into three (rather than the usual two) every hour. If there are ten of these little bugs in a petri dish at 9:00 AM in the morning, how many will there be by quitting time at 5:00 PM.

Miscellaneous Problems

1. The atmospheric pressure P (in psi) is approximated by

P = 14.7e–.1h

where h is the altitude above sea level in miles.

a. Mt. McKinley, in Alaska, is the highest point in North America. The elevation is 20,320 feet. What is the pressure at its summit? (1 mile = 5280 ft.)

b. The lowest land point in the world is the Dead Sea (Isreal-Jordan), where the elevation is 1299 feet below sea level. What is the atmospheric pressure at this point?

2. A healing law for skin wounds states that

A = A0e–.1t

where A is the number of square centimeters of unhealed skin after t days when the original area of the wound was A . How many days does it take for half of the wound to heal?

3. A law of light absorption of a medium for a beam of light passing through is given by

I = I0e–rt

where I0 is the original intensity of the beam in lumens, and I is the intensity after passing through t cm of a medium whose absorption coefficient is r. Find the intensity of a 100 lumen beam after it passes through 2.54 cm of a medium with absorption coefficient of .095.

4. A learning curve describes the rate at which a person learns certain specific tasks. If N is the number of words per minute typed by a student, then

N = 80(1 – e–.16t)

where n is the number of days of instruction. Assuming Joe is an average student, what is his typing rate after 20 days of instruction?

5. Members of a discussion group tend to be ranked exponentially by the number of times they participate in a discussion. For a group of ten, the number of times P , the nth ranked participant, takes part is given by

Pn = P1e.11(1 – n)

where P1 is the number of times the first-ranked person participates in the discussion. For each 100 times the top-ranked participant enters the discussion, how many times should the bottom-ranked person be expected to participate?

6. If an object at room temperature T0 is surrounded by air at a temperature Ta, it will gradually cool so that the temperature T is given by

T = Ta + (T0 – Ta)ekt

where the constant k depends upon the particular object being measured and t is given in appropriate time units (minutes or hours, etc.). This formula is called Newton's law of cooling. Solve the formula for the constant k.

7. You draw a tub of hot water (k = –.09 for time measured in minutes) for a bath. The water is 100F when drawn and the room is 72F. If you are called away to the phone, what is the temperature of the water 20 minutes later when you get in?

8. You take a batch of chocolate chip cookies from the oven (250F) when the room temperature is 74F. If the cookies cool for 20 minutes and k = –.095 when time is measured in minutes, what is the temperature of the cookies?

9. It is known that the temperature of a given object fell from 120F to 70F in an hour when placed in 20F air. What was the temperature of the object after 30 minutes?

10. An object is initially 100F. In air of 40F it cools to 45F in 20 minutes.

a. What is its temperature in 30 minutes?

b. How long will it take the object to cool to 40F?

c. How long will it take this object to cool to 75F?

11. The police discover the body of a math professor. Critical to solving the cirme is determining when the murder was committed. The police call the coroner, who arrrives at 12:00 P.M. The coroner immediately takes the temperature of the body and finds it to be 94.6. The coroner takes the temperature 1 hour later and finds it to be 93.4. The temperature of the room is 70. When was the murder committed? Assume that normal body temperature is 98.6

12. Assume that in the previous problem the body was found at 1:30 P.M. and the temperature measured at 90. The temperature was then checked at 3:30 P.M. and found to be 89.5. If the ambient room temperature is 72, when did death occur?

The logistics equation Q(t) = is a growth equation used to describe how a population quantity Q grows over time t. In this equation M represents the maximum population that a particular environment will support.

13. A population of 200 birds is introduced into an environment which will support 800 birds. After 6 weeks the population of birds has grown to 450. How long will it take to reach a population of 600 birds?

14. A population of Northern White Bears consists of 60 bears. Population growth is encouraged and after 3 years the population expands to 75 bears. If the environment will only support 1200 bears, how long will it take the population to reach 1000 bears?

15. An artificial laboratory environment will support 1,000,000 dung flys. At the beginning of an experiment 200 flys are introduced into the environment. After 30 minutes the number of flys has increased to 6000 flys. How long will it take fly population to increase to 800,000 flys?

16. The forest service has determined that Black lake is capable of supporting a population of 8000 fish and that the population needs to be at least 6000 fish before the lake can be opened up for fishing. At the beginning of a survey period it is estimated that there are 800 fish in the lake. Two years later the population grows to 1275 fish. How long will it take until the lake can be opened up for fishing?

17. A desert environment is capable of supporting a population of 15000 rock lizards. Environmentalists surveying the desert estimate a population of 1875 lizards. Five years later a second survey estimates a population of 9800 lizards. How long will it take until the population is estimated to be 12000 lizards?

18. An environment the will support 300 lemus is observed to have 50 lemus in residence. Three years later the population grows to 80 lemus. How many years will it take until the population reaches 130 lemus?

19. The jungles of Soporphia have enough water to support a population of 5000 white tailed lynx. A survey team investigating the mating habits of the lynx find that the population is 200 lynx. Ten years later the population has grown to 900 lynx. How long will it take the population to reach 2500 lynx?

20. A genetics experimenter creates an artificial environment that will support 103500 blood flies. He puts 2300 of the flies in the environment and observes their growth patterns. After 25 weeks the population of flies grows to 53000 flies. How long will it take the population to grow to 85000 flies?

21. The mountains of Kalahlalawai have enough food and water to support a population of 6080 llama. A National Geographic documentary team counts 380 llama in the mountains at the beginning of their project. 5 years later the population has grown to 1145 llama. How many years will it be until the population of llama reaches 2200?

22. The Siberian wasteland is capable of supporting 4384 snow lizards. An initial survey determines that the population is 137 lizards. Six years later the population reaches 260 snow lizards. Howl long until the population of snow lizards reaches 2100?

Answers

Interest Rate Problems

1. a. 740.12; 745.91 b. 804.06; 809.92 c. 950.08; 953.44

d. 1315.93; 1323.13 e. 1947.46; 1960.39 f. 1979.22; 1984.69

g. 1209.52; 1214.87 h. 1528.02; 1535.27 i. 1632.96; 1641.01

j. 2357.00; 2362.25 k. 2694.70; 2699.72 l. 3219.23; 3224.99

m. 7152.17; 7190.52 n. 3722.16; 3729.56 o. 3125.07; 3132.83

2. 1640.70 3. 2868.41 4. 2227.41 5. 1726.16

6. 1507.08 7. 13.02 8. 17.75 9. 8.31

10. 17.67 11. 17.42 12. 23.28 13. 2001.60

14. 2009.66 15. 2288.98 16. 6386.12 17. 13742.19

18. 28240.43 19. 12.02; 3.96 20. 3823.98 21. 101.68

22. 1016736.11 23. 8.84 24. 8.66 25. 10285.33

26. 10724.94 27. 7.25% 28. 3487.5 29. 1144.85

30. quarterly 31. 71,490,693 32. 472.76

Growth and Decay Problems

1. 97.29 2. 15.47 3. 9.7 4. 1.14

5. .354 6. .886 7. 30 8. 28.17

9. 10.81 10. 626469 11. 275006 12. 174470

13. 156859 14. 14050712 15. 17161578 16. 9,742,443

17. 22076019 18. 2963039; 3394700; 3889246

19. 576410; 1050288; 1913750 20. 220.84 21. 960.79

22. 11.6 23. 5.33 24. –5.33 1010 25. 69; 138.6

26. –.000124 27. 11180 28. 10266 29. 17099

30. 1311 31. 8700 32. 57% 33. 15300

34. 5430 35. 11994 36. 2616 37. 186

38. 4.4 39. 9921 40. 19932 41. 3:10

42. 33.5 43. 125.75 44. 1.66 45. 9

46. 65610

Misc. Problems

1. a. 10.004; b. 15.066 2. 7 3. 78.56

4. 77 5. 37 6. k =

7. 76.6 8. 100.32 9. 90.7

10. a. 41.44; b. never; c. 4.34 min

11. k = –.05, t = 3.013 hours, murder took place at 9:00

12. k = –.014, t = 27.7 hours

13. k = .225, t = 9.77 14. k = .0788, t = 57.79 15. k = .1136, t = 87.2

16. k = .267, t = 12.34 17. k = .516, t = 6.5 18. k = .199, t = 6.73

19. k = .166, t = 19.125 20. k = .1533, t = 34.63 21. k = .249, t = 8.58

22. k = .11168, t = 30

3/24/98

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