Math 1010
Drug Filtering Lab
Name:_____________
The purpose of this lab is to come up with a continuous model for exponential decay.
Dot assumes that her kidneys can filter out 25% of a drug in her blood every 4 hours. She knows
that she will need to take a drug test for an interview in a couple of days. She plans on taking one
1000-milligram dose of the drug to help manage her pain.
1.) Fill in the table showing the amount of the drug in your blood as a function of time and round
each value to the nearest milligram. The first two data points are already completed.
TIME SINCE
TAKING
THE DRUG (HR)
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
What might a model for this data look like?
AMOUNT OF
DRUG
IN HER BLOOD
(MG)
1000
750
563
422
317
238
179
134
101
76
57
43
32
24
18
14
11
8
2.) Use a graphing utility to make a plot of the above data. Label axes appropriately.
1000
900
800
700
600
Milligrams of
500
Drug
400
Amount of Drug in Blood
Time Since Taking the Drug
300
200
100
0
0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68
Hours Since Taking Drug
3.) Based on your graph, what can you say about the data? For example, is there a pattern? Is
there constant slope?
4.) How many milligrams of the drug are in Dot’s blood after 2 days?
32MG
5.) How many milligrams of the drug are in Dot’s blood after 5 days?
0 MG
6.) How many milligrams of the drug are in Dot’s blood 30 hours after she took the drug? Explain
your reasoning.
28 hours= 134 MG, 32 hours= 101 MG. 30 hours= 117/118MG [134*1/2(.25)-1] or 134*.875.
Half of 25% is 12.5%, which represents 2 hours instead of 4 hours.
7.) A blood test is able to detect the presence of this drug if there is at least 0.1 mg in a person’s
blood. How many days will it take before the test will come back negative? Explain your
answer.
Due to the rate of 25% decrease every 4 hours, the drop becomes less and less significant
over time. At this rate, it would take 172 Hours, or just over 7 days, for the drug to be 0.00MG
or less. At 140 hours, the drug would be at 0.04MG, which is less than 0.1MG, which is just
shy of 6 days.
8.) Will the drug ever be completely removed from her system? Explain your reasoning. What
complications might arise from having excess amounts in her system?
The drug, at some point would be completely removed from her system, but trace amounts
remain for a long period of time, even if the trace amount is undetectable. For example, at 424
hours, or almost 17 ½ days, the trace amounts are 0.00000000001MG which is almost
nonexistent, but still remains. By increasing the initial MG amount, it would take even longer to
exit the system. The problem is that 25% of a the MG remaining after 4 hours keeps getting
smaller, but even 25% of 0.00000000001 still produces a positive number. The decrease is
less and less over time.
9.) Since there is a constant rate of decay, a continuous exponential decay model can be used to
determine how much drug is in her system at any time.
Exponential Decay Model
A(t )  A0e kt
Where A(t) is amount of drug in blood at time t in hours,
A0 is the initial amount of drug, and
k is the rate of decay (it will be a negative number)
You will have to find the actual value of k that works for this model. Write down the
exponential decay model for the amount of drug in Dot’s blood as a function of time:
Model: A(t)=1000mg*e^t[(1/4)*ln(3/4)] Example: 750=1000e^4[(1/4)*ln(3/4)]
A=1000mg, T=Hours (0) at first, K=Rate of drug leaving the body
Now use that model to fill in the following table:
TIME SINCE
TAKING
THE DRUG (HR)
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
AMOUNT OF
DRUG
IN HER BLOOD
(MG)
1000
750
563
422
316
237
178
133
100
75
56
42
32
24
18
13
10
8
10.)
Interpret the parameters of this exponential model in terms of the context of the
problem.
The rate of decay (k) is -0.0719 or about 7.2%. The drop in dosage becomes less and less as
the drug continues to leave the system. BY solving for k, we can determine a consistent decay
rate based off the given time (t).
11.)
Compare your values with the estimated values in the model. How close were they?
Why might they be different?
The numbers were very close, only differing by a +/- a couple MGs. They are different because
the formula gives a more exact method of calculating how long it takes the drug to exit the
system.
12.)
Use a graphing utility to graph the original data along with a graph of the model on the
same set of axes.
1000
900
800
700
600
Milligrams of
500
Drug
400
Amount of Drug in Blood
Time Since Taking the Drug
300
200
100
0
0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68
Hours Since Taking Drug
1200
Rate of Decay Formula Chart
1000
800
Amount of Drug in Blood
600
Time Since Taking the Drug
400
200
0
0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68
13.)
Were you expecting a horizontal asymptote? What might that mean in the context of
the problem?
The result of a horizontal asymptote could be because of the rate the drug enters the system
as well as the drug leaving the system will cause the line to cross the X-axis.
14.)
Using your model, how much drug is in her system 17 hours after taking the drug?
A(t)=1000e^17[(1/40*ln(3/4)] = 294mg
15.)
Using your model, how long will it take for exactly one-half of the drug to remain in her
system?
9.64 Hours, the remaining drug is 499.9, or 500mg: 500=1000e^9.64[(1/40*ln(3/4)]
16.)
Using this model, how long will it take for 0.1 mg of the drug to remain in her system?
128 Hours, the amount of the drug remaining is 0.1mg.
17.)
Do you think the continuous decay model is more accurate for predicting the amount of
drug in her blood? Why? Or why not?
Yes, by using this formula, you can create a more accurate rate for the drug leaving the body,
even if the difference is minimal, that difference can still be the reason she fails or passes her
drug test.
18.)
What other factors should be considered in coming up with a more realistic model?
Additional factors can be considered when finding the rate of decay, such as body weight,
body fat percentage, metabolism, age, gender, and what drug specifically a person is taking
because some drugs have a longer half-life than others and can remain in your system
exponentially longer.
19.)
Reflective writing: Did this project change the way you think about how math can be
applied to the real world? Write one paragraph stating what ideas changed and why. If this
project did not change the way you think, write how this project gave further evidence to
support your existing opinion about applying math. Be specific.
This project helped me understand how a formula can be constructed to determine the
outcome or estimated results of a given situation. Whether the math is used to determine the
amount of a drug in your system, or to determine the amount of money you can gain from a
compound interest bank account, numbers never lie and they are always applicable. I continue
to realize how math is involved in almost everything we do on a daily basis. It’s how we can
understand worldwide statistics in the world, health industry, sports, or business industry.
Before this project, I never understood how a function could determine the life of a drug in your
system, and it is very interesting.