For a certain procedure, a cat is anesthetized (drugs are eliminated exponentially from the blood stream). Of any drug in the blood stream, the body will eliminate half in 3 hours. If it takes 30 milligrams of the drug to anesthetize the cat, what single dose is needed for a 1 hour operation?

Question

For a certain procedure, a cat is anesthetized (drugs are eliminated exponentially from the blood stream).  Of any drug in the blood stream, the body will eliminate half in 3 hours.  If it takes 30 milligrams of the drug to anesthetize the cat, what single dose is needed for a 1 hour operation?

anonymous · Answer

@amistre64

anonymous · Answer

If in 3 hours, half the drug is eliminated, then the whole thing will be completed in 6 hours.
If 30 ml of drug is required to anesthetize the cat in 6hrs, so complete the process in 1 hr, the dosage should be equivalent. So, i think it should be 6*30=180 ml.

anonymous · Answer

finally got back on. it turned out to not be 180 :/

kainui · Answer

That's because Sourav was thinking linearly when the drug really decreases in concentration exponentially. Since we know it's exponentially decaying, what do you think the function will look like?

anonymous · Answer

not sure actually

kainui · Answer

$$Q(t)=Q_0 e^{-at}$$ This is what exponential decay looks like. Q(t) represents the concentration Q as a function of time t. Qo is the initial amount, and you can prove that to yourself by plugging in t=0 to this formula. The negative sign shows that it is indeed decaying exponentially.

They say that for any amount you put in, you'll have exactly half left after 3 hours. So we can just plug that in: $$\frac{Q_0}{2}=Q_0e^{-a3}$$ and use this to solve for our rate constant, a.

Then next we need to make sure that by the end of the operation that we have 30 mg of concentration. So that means our final amount has to be 30 since the cat's blood metabolizes it over time and we need to put in more. So to calculate the amount we need to put in initially, we plug that final concentration in along with the time it takes to get there.

Q(1)=30

is what that is in symbols since after 1 hour we should have 30 mg left since the procedure is over we don't want it to be higher than that.

anonymous · Answer

oh so it comes out to be 30?

kainui · Answer

Yeah, you would want to inject more than 30 mg at the beginning of the procedure, otherwise the cat will wake up! You have to have at least 30 mg in the cat's bloodstream during the course of the entire operation.

anonymous · Answer

that's what the answer is though. i just saw that it was in the question

anonymous · Answer

isn't* sorry

kainui · Answer

Read what I wrote, think about it, and try to understand what's going on. I'll be back in about an hour.

anonymous · Answer

Back

anonymous · Answer

Yeah it wasn't 30 :/

kainui · Answer

30 is the final concentration. But your answer isn't the final concentration. You are solving for the initial concentration.

anonymous · Answer

oh shoot. how do you proceed. sorry i didn't notice. -.-

kainui · Answer

I've written out the whole procedure in that longer post above. If you have trouble understanding it just ask and I'll help you understand.

anonymous · Answer

Ok. i need help please