Unformatted text preview: Kecana
In all of the following problems assume exponen-
in 0.5 h. However, bacteria are continuously
1. One estimate of the growth rate of the United
siphoned off at a rate of 5 g/h. Initially, there
States is 1.5% per year. How many years will it
are 10 g of bacteria. How much bacteria are
take for the population to double?
there in the vat after 2 h?
2. A crystal grows by 5% in one day. When would
11. The rate of interest at one bank is 3% per
you expect the crystal to be twice its original
year, whereas the yield at another bank is 3%
a year. Both offer continuous compounding.
Find the two doubling times.
3. A certain bacterium is observed to double in
8 h. What is its growth rate?
12. The cost of a two-liter bottle of your favorite
4. The population of the world is expected to
soft drink was 85 cents two years ago, but it
double in the next 30 yr. What is its growth
now costs 95 cents. If this rate of increase
continues, approximately when will the bottle
5. The growth rate of a certain strain of bacteria
is unknown, but assumed to be constant. When
13. The GDP (gross domestic product) of a certain
an experiment started, it was estimated that
country increased by 6.4% during the last year.
there were about 1500 bacteria, and an hour
If it continued to increase at that rate, approxi-
later there were 2000. How many bacteria
mately how many years would it take for the
GDP to double?
would you predict there are 4 h after the
14. During one year, food prices increased by 15%.
6. A population of bacteria is initially given and
At that rate, in approximately how many years
grows at a constant rate k. Suppose + hours
would food prices triple?
later the bacteria are put into a different cul-
15. Cost of living is the amount required to pur-
ture such that the population now grows at the
chase a certain fixed list of goods and services
constant rate k2. Determine the population of
in a single year. We assume that it is subject to
bacteria for all time.
exponential growth. The growth rate is known
7. The doubling time for a certain virus is 3 yr.
as the rate of inflation. If the cost of living rose
How long will it take for the virus to increase
from $10,000 to $11,000 in one year (a 10% net
to 10 times its current population level?
increase), what is the instantaneous rate of
increase in the cost of living in that year?
8. Initially you have 0.1 g of a bacteria in a large
Equivalently, what is the rate of inflation?
container; 2 h later you have 0.15 g. What is
the doubling time for these bacteria?
16. Over a 3-yr period, housing prices increased
15%. At that rate, how many years would it
9. An organism living in a pond reproduces at a
take for housing prices to increase 50%?
rate proportional to the population size. Or-
ganisms also die off at a rate proportional to
17. A radioactive isotope has a half-life of 16 days.
the population size. In addition, organisms are
You wish to have 30 g at the end of 30 days.
continuously added at a rate of k g/yr. Give
How much radioisotope should you start with?
the differential equation that models this
18. A radioisotope is going to be used in an exper-
iment. At the end of 10 days, only 5% is to be
10. A bacterium is reproducing in a large vat of
left. What should the half-life be?
nutrients according to an exponential growth
19. A radioactive isotope sits unused in your labo-
law that would cause the population to double
ratory for 10 yr, at which time it is found to...
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- Fall '10