Question

Penicillin has a half life of 36 hours in the bloodstream. Suppose that a patient recieves an initial dose of 30 milligrams of penicillin at 9am.

How much penicillin will be in the patients blood at 9pm that night?

Approximate when the penicillin concnetration will reach 1% of its original level?

What is the decay rate for penicillin?

Answer 1

I'm not always great at these kind of questions, but the rate change of the drug \(dP/dt\) is proportional to some constant \(r\) times the amount of the drug \(P\)
because it's decay it's negative...
so we can write\[\frac{dP}{dt}=-rP\]

Answer 2

the solution to that differential equation is\[P(t)=P_0e^{-rt}\]where \(P_0\) is the initial dose.
I think we should say \(t\) is in hours
now we need to find \(r\)...

Answer 3

we are given that the half life is 36 hours you say? not minutes, right?

Answer 4

Correct

Answer 5

ok then we have that the amount in the bloodstream is half when\[\frac{P_0}2=P_0e^{-36r}\]can you solve this for \(r\) ?

Answer 6

and since the intial dose \(P_0\) is known to be 30mg we can write that in if you like...\[\large P(36)=15=30e^{-36r}\]can you solve it for \(r\) ?

Answer 7

6.662463305E-54

Answer 8

that sounds wrong...

Answer 9

\[\frac{P_0}2=P_0e^{-36r}\]\[\frac12=e^{-36r}\]\[\ln\frac12=-\ln2=-36r\]\[r=\frac{\ln2}{36}\]

Answer 10

so from 9am to 9pm how many hours is that?

Answer 11

36-12=24hours

Answer 12

12 hours

Answer 13

But aside from that, I've made a mistake...
these really aren't my kind of problems, I was just trying to give you quick help
now I need to figure out what I did wrong

Answer 14

Thanks anyway...these aren't easy and I am finding difficulty imputing these formulas in my caluculator.

Answer 15

ah, you want to do it the calculator way, eh?
I see my mistake now though, the formula is...

Answer 16

Our teacher allows calculator on the quiz...

Answer 17

\[\huge P(t)=(30)2^{-\frac t{36}}\]a) plug in t=12 hours and you will have the amount in the body at that time
b) solve\[\huge\frac{30}{100}=(30)2^{-\frac t{36}}\]for t and you will find when the concentration is one percent its original
good luck!

Answer 18

oh, and c)
the decay rate is \(r\), which we found earlier

Answer 19

(30)2^-13/36?

Answer 20

from 9am to 9pm is 12 hours, so you need to put in t=12, not 13

Answer 21

(30)2^-12/36=syntax error