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
3. Cm242 has a half-life of 163 days. How much remains of 10 grams after one
4. Np239 has a half-life of 2.237 days. How much remains of 10 grams after one
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 k = –.0231?
8. What is the half-life of strontium 90 (in years) if the decay constant is k = –.0246?
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
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
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
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
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 6 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 10 5 bacteria
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.
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