Download Drug Dosage - Web4students

Math 101 – Section 5.4 – Applications of Exponential and Logarithmic Functions
1) Population Model
The population of a colony of mosquitoes is growing exponentially. Assume there are 1000
mosquitoes as of today and that each day the population doubles. Let M(t) represent the total number
of mosquitoes at t days since today.
a) Find an equation for M.
b) Is this a case of exponential growth or exponential decay?
c) What is the y-intercept of the graph? Interpret within the context of the problem.
d) What will be the number of mosquitoes in 3 days?
e) When will the population be 32,000 mosquitoes?
f) Interpret the ordered pair (1, 2000) within the context of the problem.
2) Magnitude of an Earthquake
The energy released by an earthquake is sometimes measured using the Richter scale. The Richter
number, R, of an earthquake is given by
R  log(
A
)
Ao
where A is the amplitude of the shockwave of the earthquake and Ao is
the amplitude of the shockwave of the smallest earth movement that can be recorded on a seismograph.
6
a) Find the Richter number for an earthquake with an amplitude of 3.0  10 times the reference
amplitude Ao
b) What is the amplitude of an earthquake with Richter number R = 8.6. Express the answer in terms of
Ao .
3) Population of Kenya
Based on present birth and death rates, the population of Kenya is expected to increase according to the
0.041t
formula N = 20.2 e
, with N in millions and t = 0 corresponding to 1985.
a) Is this a case of exponential growth or exponential decay?
b) What is the y-intercept of the graph? Interpret within the context of the problem.
c) What was the population in the year 1995?
d) How many years will it take for the population to double?
4) Loudness of Sound
The loudness of sound can be measured using a decibel scale. The sound level L (in decibels) of sound
I
L  10log( ) where I is the intensity of the sound (watts per square meter, W / m2 )
Io
12
2
and I o  10 W / m . The constant I o is the approximate intensity of the softest sound that can be
is given by
heard by humans.
10
a) Find the decibel readings for a whisper which intensity is 10 .
b) What is the intensity of a sound that has a loudness of 80 decibels? Write the answer in terms of I o
.
1
5) Radioactive Substance
The worst nuclear accident in the world occurred in Chernobyl, Ukraine on April 26, 1986.
Immediately after the disaster, 31 people died from acute radiation sickness. It is predicted that about
17,000 people will die from radiogenic cancer, mostly due to the release of the radioactive element
cesium-137.
t
Let P  f (t )  100(.977) represent the percent of the cesium-137 that still remains at t years since
1986.
a) Is this a case of exponential growth or exponential decay?
b) What is the y-intercept of the graph? Interpret within the context of the problem.
c) What percent of cesium is still remaining today?
d) What is the half-life of cesium-137? (Half-life is the time it takes for the quantity to be reduced to
half)
6) Earthquake area in the West
In the western United States, the area A (in square miles) affected by an earthquake is related to the
magnitude R of the quake by the formula
R = 2.3 log (A + 3000) – 5.1
a) If the area affected by an earthquake is 2000 square miles, what is the magnitude of the earthquake?
b) If an earthquake has magnitude 5 on the Richter scale, estimate the area A of the region that will
feel the quake.
7) Drug Dosage.
A drug is eliminated from the body through urine. Suppose that for a dose of 10 milligrams, the
amount A(t) remaining in the body t hours later is given by
A(t) = 10(0.8)^t
a) Is this a case of exponential growth or exponential decay?
b) What is the y-intercept of the graph? Interpret within the context of the problem.
c) Estimate the amount of the drug in the body 8 hours after the initial dose.
d) What percentage of the drug still in the body is eliminated each hour?
e) Suppose that in order for the drug to be effective, at least 2 milligrams must be in the body.
Determine when 2 milligrams is left in the body.
f) What is the half-life of the drug?
g) Graph this function and check all answers a-d
8) Children’s Growth
The Count Model is a formula that can be used to predict the height of preschool children. If h is
height (in centimeters) and t is age (in years), then
H = 70.228 + 5.104t + 9.222 ln t,
for ¼ ≤ t ≤ 6
The rate of growth R (in cm/year) is given by
R = 5.105 + 9.222/t
a) Predict the height and rate of growth of a typical 2 year old.
b) Use a graph to answer:
At what age do we expect a preschool child to have a height of 100 cm?
c) What is the rate of growth at the moment when the height is 100 cm?
2
9) Employee Productivity
Suppose a manufacturer estimates that a new employee can produce five items the first day on the job.
As the employee becomes more proficient, the daily production increases until a certain maximum
production is reached. Suppose that on the nth day on the job, the number f(n) of items produced is
approximated by
f (n)  3  20(1  e( 0.1n ) )
a) Estimate the number of items produced on the fifth day, the ninth day, the twenty-fourth day,
the thirtieth day, and the sixtieth day.
b) When will the employee be producing 8 items.
c) Sketch the graph of f from n = 0 to n = 60 (Graphs of this type are called learning curves and
are used frequently in education and psychology)
10) Language Dating
Glottochronology is a method of dating a language at a particular stage, based on the theory that over a
long period of time linguistic changes take place at a fairly constant rate. Suppose that a language
originally had N o N o basic words and that at time t, measured in millennia (1 millennium = 1000
years), the number N(t) of basic words that remain in common use is given by N(t) = N o (0.805)^t
a) Approximate the percentage of basic words lost every 1000 years
b) If N o = 200, sketch the graph of N for 0 ≤ t ≤ 5
c) After how many years are one-half the basic words still in use?
11) Compound Interest
A person invests $10,000 in an account that earns 5% interest compounded annually. The term
compounded annually means that the interest earned each year equals 5% of the sum of the $10,000
and any interest earned in previous years. Let V = f(t) represent the value (in dollars) of the account at t
years after the money is invested.
a) Find an equation for f.
b) What will be the value after 5 years?
c) How long will it take for the amount to double?
3