Modelling and Problem Solving Question

May 2010
4
0
Hello,
I've been trying to solve this problem for a while, and it's really getting to me. We're supposed to use Microsoft Excel, but I don't know where to start with the problem. It is as follows:

When the kidneys eliminate a chemical from the blood, they tend to eliminate a certain proportion each time period. For example, the average person eliminates about 13% of the caffeine in his or her body each hour. In doses larger than 20mg, caffeine can act as a mild stimulant (it can be toxic in large doses). Assume that a 375mL can of Cola has about 45mg of caffeine.
Suppose that you have a can of Cola on the hour for three hours. If you had your last Cola at 10 p.m, when can you expect the caffeine that you have ingested to stop having its effect?

I'm probably making some silly assumptions, but I'm not sure how to start. (Headbang) Thank you.
 
Last edited:

Debsta

MHF Helper
Oct 2009
1,279
584
Brisbane
What does "Suppose that you have a can of Cola on the four for three hours" mean???
 
May 2010
4
0
Sorry, I was typing it quickly:
Suppose that you have a can of Cola on the hour for three hours. If you had your last Cola at 10 p.m, when can you expect the caffeine that you have ingested to stop having its effect?
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Hello,
I've been trying to solve this problem for a while, and it's really getting to me. We're supposed to use Microsoft Excel, but I don't know where to start with the problem. It is as follows:

When the kidneys eliminate a chemical from the blood, they tend to eliminate a certain proportion each time period. For example, the average person eliminates about 13% of the caffeine in his or her body each hour.
So, letting C(t) be the amount of caffeine in your body t hours after ingesting it, \(\displaystyle \frac{dC}{dt}= -.13C\)

In doses larger than 20mg, caffeine can act as a mild stimulant (it can be toxic in large doses). Assume that a 375mL can of Cola has about 45mg of caffeine.
Suppose that you have a can of Cola on the hour for three hours. If you had your last Cola at 10 p.m, when can you expect the caffeine that you have ingested to stop having its effect?
Taking t= 0 when you drink the first can, C1(0)= 45. Solve \(\displaystyle \frac{dC1}{dt}= -.13C\) with C1(0)= 45, then find C1(1) when you drink the next can of cola. (Integrate \(\displaystyle \frac{dC1}{C1}= -.13dt\).)

At this point, \(\displaystyle \frac{dC2}{dt}= -.13C2\) and C2(0)= C1(1) (I have "reset" the clock so the time at drinking the second cola is the new t= 0). Solve for C2 and find C2(1).

Finally solve \(\displaystyle \frac{dC3}{dt}= -.13C3\) with C3(0)= C2(1).
Find t such that C3(t)= 25. Remember that this "t" is in hours after 10:00 PM, the time you had the last cola.

I'm probably making some silly assumptions, but I'm not sure how to start. (Headbang) Thank you.
 
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