Question

Carbon dioxide at high enough concentrations is lethal. According to the EPA (http://www.epa.gov/ozone/snap/fire/co2/co2report.html), exposure to ambient air concentrations greater than 17 % can cause death within one minute. Recall that we constructed the following model for gas exchange in the human lung:

ct+1=(1−q)ct+qγ,

where q is the fraction of air in the lung that is exchanged for ambient air at each breath, γ is the concentration of chemical in the ambient air, and ct is the concentration of chemical in the lung.

Answer 1

Suppose that the ambient air has a CO2 concentration of 20 % (a concentration sufficient for fire suppression), and that a fire fighter whose oxygen mask has failed is exposed to this air. Suppose that the fire fighter has no CO2 in his lungs and that he exchanges 30 % of the air in his lungs for ambient air at every breath. In the following, you can use γ=0.2 (so that ct is measured as the fraction of the air that is CO2).

Answer 2

Now suppose that the fraction of gas exchanged at each breath is only 10 %. What is the concentration in the lung after 4 breaths? so c4 = ? this is where I am stuck!!

Answer 3

First off, is this your formula?
\[C_t+1=(1-q)C_t+q \gamma\]

Answer 4

yes

Answer 5

thank you for responding!! :)

Answer 6

I found equilibrium which is C^* = 0.2

Answer 7

\(q\) is the fraction of air in the lung that is exchanged for ambient air at each breath = 0.3
\(\gamma\) is the concentration of chemical in the ambient air = 0.2
\(C_t\) is the concentration of chemical in the lung = what we're looking for

where does the 10% come in?

Answer 8

sorry I'm just laying it all out, trying to understand the question

Answer 9

the 10% is the next part of the question, so I think they are saying now the fraction of gas exchanged at each breath is only 10 %, whereas before I think it was 20%

Answer 10

No not 20%, it was 30% originally

Answer 11

"in his lungs and that he exchanges 30 % of the air in his lungs for ambient air at every breath"

Answer 12

ok, so it does look like it's asking for \(c_4\), but I'm not seeing where you'd put in the 4

Answer 13

I think I have to come up with c1 first, so I have to have a discrete time dynamical system so that I can plug in c1 and do it 4 times to get to c4. Also I am assuming that c0=1 (initial value)

Answer 14

ok, so you're supposed to do iterations. that makes sense

Answer 15

Is the 1 supposed to be in the subscript?

Answer 16

Like this?
\[C_1=(1-0.3)(0.2)+(0.3)(0.2)\]
And then use this to find \(C_2\)?

Answer 17

yes, iteration! I think you are on to something... so for C4 - let me see what I get...

Answer 18

so wait shouldn't the 0.2 be 0.1, since the concentration is now 10%?

Answer 19

I meant the 0.3 sorry not the 0.2

Answer 20

Yes it's 0.1

\(C_{t+1}=(1-q)C_t+q \gamma\)

\(C_t\) is the only thing that changes, so the equation is basically
\(C_{t+1}=(1 - 0.1)C_t+(0.1)(0.2)\)

\(C_{t+1}=0.9C_t+0.02\)

Then iterate
\(C_{1}=0.9(0.1)+0.02=0.11\)
\(C_{2}=0.9(0.11)+0.02=0.119\)
\(C_{3}=0.9(0.119)+0.02=0.1271\)
\(C_{4}=0.9(0.1271)+0.02=0.13439\)

So the concentration after 4 breaths is 13.4%.
That's what I'm thinking anyway