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Assignment #4 Exponentials and Logarithms (Show your work. NO WORK NO CREDIT.)
1. Radioactive decay and Smoke Detectors
A mass of m = 2 grams of a radioactive isotope is used by a firm
that manufactures smoke detectors. Small amounts of the
isotope are used in each detector to create ions that enhance
the detection of smoke particles. Unfortunately, the radioactive
substance decays at such a rate that only one gram is left after
six months. Find the mass m(t) remaining as a function of time it
in years. (Show your work below.)
2. Municipal Bonds A city has sold one hundred million
dollars in bonds to raise money for a redevelopment project.
The bonds mature in 10 years and pay an 8% interest
compounded annually. What will the bill be when the bonds
mature? (Show your work below.)
3. Air Pressure Air pressure, P, decreases exponentially with the height above the surface of the
earth, h:
P = P0e-0.00012h
where P0 is the air pressure at sea level and h is in meters. (Show your work below.)
(a) If you go to the top of Mount McKinley, height 6198 meters (about 20,330 feet), what is the air
pressure, as a percent of the pressure at sea level?
(b) The maximum cruising altitude of an ordinary commercial jet is around 12,000 meters (about
39,000 feet). At that height, what is the air pressure, as a percent of the sea level value?
Copyright 2015 by David R. Hill Mathematics Department Temple University
4. Under certain circumstances, the velocity, V, of a falling raindrop is given by V = V0(1 – e-t ), where
t is time and V0 is a positive constant. (Show your work below.)
(a)
Sketch a rough graph of V against t, for t ≥ 0.
(b) What does V0 represent?
5. The population of Kenya was 19.5 million in 1984 and 21.2 million in 1986. Assuming the
population increases exponentially, find a formula of the form P = P0e-kt for the population of Kenya
as a function of time. Determine the rate, as a percent, at which the population is growing. (Show
your work below.) (Hint: use t as the number of years since 1984.)
6. The air in a factory is being filtered so that the quantity of a pollutant, P (measured in mg/liter), is
decreasing according to the equation P = P0e-kt, where t represents time in hours. If 10% of the
pollution is removed in the first five hours: (Show your work below.)
(a) What percentage of the pollution is left after 10 hours?
(b) How long will it take before the pollution is reduced by 50%?
Copyright 2015 by David R. Hill Mathematics Department Temple University
7. The release of chlorofluorocarbons used in air conditioners and, to a lesser extent, in household
sprays (hair spray, shaving cream, etc.) destroys the ozone in the upper atmosphere. At the present
time, the amount of ozone, Q, is decaying exponentially at a continuous rate of 0.25% per year. What
is the half-life of ozone? In other words, at this rate, how long will it take for half the ozone to
disappear? (Show your work below.)
Copyright 2015 by David R. Hill Mathematics Department Temple University