Exponential & Logarithmic Word Problems
1. If bacteria in a certain culture double every half hour, write an equation that gives the
number of bacteria N in the culture after t hours, assuming the culture has 100 bacteria
at the start.
a) How much is present at 2 hours?
b) Graph the equation for .
2. The radioactive isotope technetium 99m is used in imaging the brain. The
isotope has a half-life of 6 hours. If 12 milligrams are used, how much will be present
aftera) 3 hours?
b) 24 hours?Compute your answers to three decimal places.
3. A couple just had a new child. How much should they invest now at 8.25%
compounded daily in order to have $40,000 for the child’s education 17 years from now? Compute your answer to the nearest dollar.
4. If $4000 is deposited into an account earning 9% compounded weekly and, at the same
time, $6000 is deposited into an account earning 7% compounded weekly, will the first
account be worth more that the second? If so, when?(Solve graphically)
5. In 1996 the population of Germany was 84 million and the population of Egypt was
64 million. If the populations of Germany and Egypt grow continuously at annual rates
of % and 1.9 % respectively, in what year will Egypt have a greater population
than Germany?(Solve graphically)
6. As long as a plant or animal is alive, Carbon 14 is maintained in a constant amount in
its tissues. Once dead, however, the plant or animal ceases taking in carbon, and carbon 14 diminishes by radioactive decay according to the equation , where A is the amount after t years and is the amount when . Estimate the age of a skull uncovered in an archaeological site if 10% of the original amount of carbon 14 is still present. Compute your answers to three decimal places.
7. Using the information in problem 6, what is the half-life of Carbon 14?
Compute your answer to the nearest year.
8. The concentration of alcohol in a person’s blood is measurable. Recent medical
research suggests that the risk R (given as a percent) of having an accident while
driving a car can be modeled by the equation where x is the variable
concentration of alcohol in the blood and k is a constant.
a) Suppose that the concentration of alcohol in the blood of 0.04 results in
a 10% risk (R=10) of an accident. Find the constant k in the equation.
b) Using this value of k, what is the risk if the concentration is 0.17?
c) Using the same value of k, what concentration of alcohol corresponds to
a risk of 100% ?
d) If the law asserts that anyone with a risk of having an accident of 20% or more
should not have driving privileges, at what concentration of alcohol in the
blood should a driver be arrested and charged with a DUI?
9. The pH of a chemical solution is given by the formula , where is
the concentration of hydrogen ions in moles per liter. Values of pH range from 0 (acidic)
to 14 (alkaline).
a) Find the pH of a 1 liter container of water with 0.0000001 mole of hydrogen ion.
b) Find the hydrogen ion concentration of a mildly acidic solution with a pH of 4.2.
10. A fossilized leaf contains 70% of its normal amount of carbon 14.
a) Solve for k using 5600 years as the half-life of the carbon 14.
b) How old is the fossil?
11. The number of watts w provided by a space satellite’s power supply after a period of
d days is given by the function .
a) How much power will be available after 30 days ?
b) How much power will be available after 1 year ?
c) How long will it take for the available power to drop to 30 watts ?
d) How long will it take for the available power to drop to 5 watts ?
12. $ 400 is invested in an account at 7% for three years.
a) How much is earned if it is compounded yearly ?
b) How much is earned if it is compounded monthly ?
c) How much is earned if it is compounded quarterly ?
d) How much is earned if it is compounded continuously ?
e) How long will it take to double compounded continuously ?
f) How long will it take to triple compounded continuously ?
13. Newton’s Law of Cooling states that the temperature u of a heated object at a given
time t can be modeled by the following function , where T
is the constant temperature of the surrounding medium, is the initial temperature of
the heated object and k is a negative constant. An object is heated to 100C and is then
allowed to cool in a room whose air temperature is 30C.
a) If the temperature of the object is 80C after 5 minutes, when will it reach 50C ?
b) What will the temperature be after 39.22 minutes?
14. It has been stated that the threshold of pain for humans is 120. Being an old man
and having hearing loss over the years of rockin’ on, Mr. Christensen’s threshold of
pain is 135 . What is the ratio of the intensity of Mr. Christensen’s threshold of
pain compared to the “normal” person ? Recall that,
15. The population of beavers at a nature preserve follows a logistic model with an initial
population of 200 beavers in 1991. Given the following information, use your
graphing calculator to answer the following questions.
a) Sketch a graph of the function from 1991 to 2006.
b) How many beavers were there in 1995 ?
c) During what year did the population first exceed 1300 ?