00:01
Hi there.
00:02
So for this problem, we're told that the concentration of a drug in the body decreases exponentially after a dosage is given.
00:10
In one clinical study, otto suggests average 14 micrograms per milliliter, okay? of the drug interblood plasma, one hour after a 100 milligrams dosage.
00:34
And three milligrams after four hours after dosage so let me just put that in here so we will have 1 ,000 milligrams then yes okay so this is the concentration after one hour and this after a 1 ,000 milligrams and 3 and so then we will have 3 milligrams grams per milliliter, and this is after four hours, okay? so that is the information that we are given.
01:22
Now, for part a of this problem, we are asked about to find the value of k and write an equation for an exponential function that can be used to predict the concentration of the drug at any given time, okay? so, first of all, for part a, we are told that initial concentration times the exponential of minus the concept of proportionality is equal to 14 this at the first hour.
01:55
And then after four hours, we will have this is equal to three.
02:06
Okay.
02:08
So with this set, what we can do in here is, well, we can take the ratio between these two expressions.
02:21
Okay, we can cancel this with this.
02:23
Then we will have the exponential of minus k divided by the esponemential of minus three times k is equal to 14 divided by three.
02:32
Then we will have that this is equal to.
02:35
So we'll have the exponential of minus k plus three times k is equal to 14 divided by three.
02:44
Then in here we will have that this is the exponential of two times k.
02:52
Okay, and then this is equal to 14 divided by 3.
02:57
So finally, solving for the constant of proportionality, we can apply the neparian logarithm in both sides of this.
03:04
So we will state that the concept of proportionality is 1 divided by 2 times the neparion logarithm of 14 divided by 3.
03:12
So that's the value for this.
03:15
So i'm not told with how many decimal places.
03:23
Okay, i'm going to, let me see, what is the solution for this? so i'm going to give you two decimal places that will be 0 .77, okay? so then this will be that the concentration at any given time is equal to the initial concentration.
03:45
Let's label this as c -0.
03:50
And then this times the exponential of the minus 0 .77, the value that we obtain times the time.
04:23
So now let's evaluate this up 0.
04:28
Well, let's use one of the conditions to obtain c -0.
04:33
So we know that after one hour is 14 microgams per millimeter.
04:37
So that will be c -0 times the espionumption of minus 0 .77.
04:41
Then we can pass this term to this size.
04:44
So the initial concentration for this is 14 divided by the s penumptial of minus 0 .77.
04:51
Then using our calculator, we obtain a value of 30 .24.
05:07
So the concentration at any given time is then 30 .24 times the exponential of minus 0 .77 times the exponential of minus 0 .77 times the time.
05:18
So that's a solution for the first question of this problem.
05:22
Now for the next question, we need to estimate the concentration of the drug after three hours.
05:27
So we use the solution that we obtained from before and evaluate that at three.
05:32
So that will be 30 .24.
05:35
And this times the exponential of minus 0 .77 times 3.
05:45
So then that will be minus 0 .77 times 3...