Download Discrete Chapter 10 Worksheet on Exponential

Name __________________________________________________ Date _________________________ Period ___________
Discrete Math Chapter 10 Worksheet on Exponential Growth and Decay
1. Suppose that after t hours there are P(t) cells present in a culture, where P ( t ) = 5000e
a. How many cells were present initially?
b. When will 20,000 cells be present?
2. The size of a certain insect population is given by P ( t ) = 300e
what time will the population equal 600? 1200?
.01t
2t
, where t is measured in days. At
3. The growth rate of a certain bacteria culture is proportional to its size. If the bacteria culture doubles in
size every 20 minutes, how long will it take for the culture to increase twelve-fold?
4. Mexico City is expected to become the most heavily populated city in the world by the end of this
century. At the beginning of 1990, 20.2 million people lived in the metropolitan area of Mexico City,
and the population was growing exponentially with a growth constant of .032 (Part of the population is
due to immigration.)
a. If this trend continues, how large will the population be in the year 2000?
b. In what year will the 1990 population have doubled?
5. Five grams of a certain radioactive material decays to 3 grams in 1 year. After how many years will just
1 gram remain?
6. One hundred grams of a radioactive substance with a decay constant of .01 is buried in the ground.
Assume that time is measured in years.
a. Give the formula for the amount remaining after t years.
b. How much will remain after 30 years?
c. What is the half-life of this radioactive substance?
7. The decay constant for cesium 137 is .023 when time is measured in years. Find the half-life of cesium
137.
Name __________________________________________________ Date _________________________ Period ___________
8. Radioactive cobalt 60 has a half-life of 5.3 years.
a. Find the decay constant of cobalt 60.
b. If the initial amount of cobalt 60 is 10 grams, how much will be present after 2 years?
9. A 4500-year-old wooden chest was found in the tomb of the twenty-fifth century B.C. Chaldean king
Meskalumdug of Ur. What percentage of the original C14 would you expect to find in the wooden
chest? (Recall that the decay constant for C14 is .00012)
10. In 1947, a cave with beautiful prehistoric wall paintings was discovered in Lascaux. France. Some
charcoal found in the cave contained 20% of the C14 expected in living trees. How old are the Lascaux
cave paintings?
11. A drought in the African veldt causes the death of much of the animal population. A typical herd of
wildebeests suffers a death rate proportional to its size.
a. Find a formula for the herd’s population at time t months.
b. How long will it take for the herd to diminish to one-tenth its original size?
12. Many scientists believe that there have been four ice ages in the past one-million years. Before the
technique of carbon dating was known, geologists erroneously believed that the retreat of the Fourth Ice
Age was about 25,000 years ago. In 1950, logs from ancient spruce trees were found under glacial debris
near Two Creeks, Wisconsin. Geologists determined that these trees had been crushed by the advance of
ice during the Fourth Ice Age. Wood from the spruce trees contained 27% of the level of C14 found in
living trees. Approximately how long ago did the Fourth Ice Age actually occur?
13. 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 for strontium is 28 years.
14. **A common infection of the urinary tract in humans is caused by the bacterium E. coli. The infection is
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generally noticed when the bacteria colony reaches a population of about 10 . The colony doubles in
size every 20 minutes. When a full bladder is emptied, about 90% of the bacteria are eliminated.
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Suppose that at the beginning of a certain time period, a person’s bladder and urinary tract contain 10
E. coli bacteria. During an interval of T minutes, the person drinks enough liquid to fill the bladder.
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Find the value of T such that if the bladder is emptied after T minutes, about 10 bacteria will remain.
(Note: the average bladder holds about 1 liter of urine. It is seldom possible to eliminate an E-coli
infection by diuresis without drugs – such as by drinking large amounts of water.)
15. **According to legend, in the fifth century, King Arthur and his knights sat at a huge round table. A
round table alleged to have belonged to King Arthur was found at Winchester Castle in England. In 1976
carbon dating revealed the amount of radiocarbon in the table to be 91% of the radiocarbon present in
living wood. Could the table possibly have belonged to King Arthur? Why?