Activity
1) Go over the
first three problems in yesterday’s
homework. Ask the students if they needed to know the last
term of the finite series to be able to calculate the sum.
(They did not need it.)
Ask if anybody came up with a reallife application where
they need to be able to find the sum of a finite geometric
series. Discuss any answers you might get.
2)
Tell the students that one application is in taking antibiotics
when they
have an infection. Let’s suppose that they
have strep throat and the doctor tells them to take the
antibiotic ampicillin regularly for two weeks. Ampicillin
is usually taken in 250 mg doses four times a day, that
is exactly every six hours. It is known that at the end
of six hours, due to excretion, about 4% of the drug is
still in the body. How much ampicillin is in the body right
after taking the eighth tablet? What about right after
the 48th tablet?
3) Ask the students how they can set up a finite geometric
series to represent the situation. Tell them to let the
symbol
represent the quantity in milligrams of ampicillin in the
blood right after having taken the nth tablet. This means
that right after taking the first tablet,
Six hours later, a second tablet is taken, and the amount
of the drug in the body becomes
because you took a new tablet (250 mg), plus you have 4%
of the first tablet as well.
Six hours later, after the third tablet is taken, the
amount becomes
because you took a new tablet (250 mg), plus you have 4%
of the second tablet, and (4%)(4%) of the first tablet as
well.
4) The students will hopefully see the pattern. Ask them
to give you the next three quantities.
Teacher Cheat sheet:
Point out that
the equations are the expanded form of a finite geometric
series with a = 250, r = 0.04, and the
highest exponent used is (n –1). There are n terms
in the series, one for each tablet.
5) Ask the students
to write the expanded form of the finite geometric series
that represents the amount of ampicillin
in the blood right after having taken the eighth tablet
and the 48th tablet, using “+…+”, instead
of writing out all of the terms.
6) Ask the students if they can think of a short cut for
calculating the amount of ampicillin in the blood right
after having taken the 48th tablet instead of writing out
all 48 terms and adding them together. Hopefully somebody
will remember how to calculate the sum using the closed
form:
7) Have the students use the closed form to calculate
the following amounts of the drug
8) Ask them to describe, as the number of doses increases,
what is happening to the amount of the drug in the body
right after a tablet is taken. Have them discuss why this
might be happening.
Teacher cheat sheet: Below are answers to the respective
amounts.
Notice that all of these values would round
to 260.42mg, the same amount (to two decimal places) as
the amount of the drug in the body right after the 48th
tablet. This means that by the end of the first day (after
taking four tablets), the amount of the drug appears to
have stabilized at approximately 260.42mg.
9) Ask the students write the closed form
if n tablets are taken. What does this closed form predict
about the longrun level of ampicillin in the body? In
other words, what happens to the level of the drug right
after a tablet is taken if a very large number of tablets
are taken?
Teacher cheat sheet: The closed form of the series for
n tablets is:
If
you allow the value of n to get very large,
Thus, assuming that 250mg continue to be
taken every six hours, in the long run the level of the
drug in the body, right after a tablet is taken, appears
to be approaching a constant level of 260.42mg.
10)
Ask the students how they can set up a finite geometric
series to represent the amount of the
drug in the body six hours after having taken a tablet.
Remind them that about 4% of the original amount is left
in the body six hours after the tablet is taken. Let the
symbol represent the quantity in milligrams of ampicillin
in the blood six hours after having taken the nth
tablet.
This means that six hours after taking the first tablet,
or just before taking the 2nd tablet,
Six hours after the second tablet is taken,
and the amount of the drug in the body becomes
as
4% of the second tablet and (4%)(4%) of the first tablet
remain in the body.
Six hours after the third tablet is taken, the amount
becomes
as there is 4% of the third tablet, (4%)(4%)
of the second tablet, and (4%)(4%)(4%) of the first tablet
remain in the body.
11)
The students will hopefully see the pattern. Ask them
to give you the next three quantities.
Teacher Cheat sheet:
Point out that the equations are the expanded form of
a finite geometric series with a = 250(0.04), r = 0.04,
and the highest exponent used is n. There are n terms in
the series, one for what remains of each tablet six hours
after it was taken.
12)
Ask the students to write the expanded form of the finite
geometric
series
that represents the amount of ampicillin
in the blood six hours after having taken the eighth tablet
or the 48th tablet, using “+…+”, instead
of writing out all of the terms.
13) Ask the students if they can think of a short cut
for calculating the amount of ampicillin in the blood six
hours after having taken the 48th tablet instead of writing
out all 48 terms and adding them together. Hopefully somebody
will remember how to calculate the sum using the closed
form:
14) Have the students use the closed form to calculate
the following amounts of the drug
15)
Ask them to describe, as the number of doses increases,
what is happening to the amount of the
drug in the body right after a tablet is taken. Have them
discuss why this might be happening.
Teacher cheat sheet: Below are answers to
the respective amounts.
Notice that all of these values would round
to 10.42mg, the same amount (to two decimal places) as
the amount of the drug in the body six hours after the
48th tablet.
16) Ask the students write the closed form six hours after
the nth tablet is taken. What
does this closed form predict about the longrun level
of ampicillin in the body six
hours after each tablet is taken?
Teacher cheat sheet: The closed form of the series for
six hours after the nth tablet is:
If
you allow the value of n to get very large,
Thus, assuming that 250mg continue to be
taken every six hours, in the long run the level of the
drug in the body, six hours after a tablet is taken, appears
to be approaching a constant level of 10.42mg.
17)
Have the students make a graph of amount of drug in the
body versus time (in six hours intervals).
Assume that the first dose is taken at time t = 0.
Teacher
Cheat Sheet: the first seven points would be as follows:
All the rest of the yvalues for Q will be 260.42mg and
all the rest of the yvalues for P will be 10.42mg.
18) Have the students make a graph of amount of drug in
the body versus number of doses. Assume that no drug
is in the body at time t = 0.
Teacher
Cheat Sheet: the first seven points would be as follows:
19) Looking at the graph of amount of
drug versus dose number, ask the students what would
happen
if they forgot
to take a tablet and went 12 hours in between doses? What
would happen if they stopped taking the antibiotic before
completing the full regimen they were given?
Teacher
Cheat Sheet: See teacher background.
Homework
The drug erythromycin may also be taken for streptococcal
infections. Erythromycin is given in 400mg doses every
six hours. At the end of six hours, there is 6.25%
of the amount taken left in the body.
1) Write the expanded finite geometric series for the
amount of the drug in the body right after having taken
the nth tablet.
2) Write the
closed finite geometric series for the amount of the
drug in the body right after having taken the nth tablet.
3) Write the
expanded finite geometric series for the amount of the
drug in the body six hours after having taken
the nth tablet.
4) Write the
closed finite geometric series for the amount of the
drug in the body right after having taken the nth tablet.
5) Calculate
the amount of erythromycin in the body right after taking
a tablet over the longrun.
6) Calculate the amount of erythromycin in the body six
hours after taking a tablet over the longrun.
7) What is the difference between the numbers calculated
in problems 5 and 6? Why does this occur?
Answer: 426.6726.67=400mg, which is the amount of each
individual dose of erythromycin taken. This occurs because
once the level of the drug in the body right after a dose
is taken reaches 426.67mg, the steady state amount, the
body excretes 400mg over six hours, (426.67)(.9375) = 400,
because taking one more tablet raises the level back to
the steady state amount.
