Ch 4 Question 5, Instructor-created question HW Score: 0%, 0...

Ch 4 Question 5, Instructor-created question HW Score: 0%, 0 of 10 points Points: 0 of 1 A single IV bolus dose of a certain drug is administered intravenously to a patient. The concentration of drug in the bloodstream \(t\) hours after the dose is administered is found to be \(C(t) = 310e^{-0.23t}\) mcg/mL. Assume this dose is administered every 4 hours. (a) What percent of the original concentration remains before the next dose is administered? (b) What is the concentration 2 hours after the second dose is administered? (c) What is the concentration 22 hours after the first dose is administered? (Hint: First, determine how many doses have been determined.) Enter the percent value with one decimal place and the C values as integers. (a) % (b) mcg/mL (c) mcg/mL Save

Transcript

00:01 All right, greetings and hello.

00:01 In this question we're told that we have the concentration of a drug in a patient's bloodstream, and it's given by this equation c of t equals 20t times e to the negative 0 .06t.

00:12 And we're asked in part a, how long does it take for the drug to reach peak concentration and what is that peak concentration? so we're going to have some function here and we want to see what the peak concentration is.

00:22 Essentially we just want to maximize this function.

00:24 So to do that we're going to go ahead and take a derivative here.

00:27 We're going to have product rule because we have two terms here.

00:30 So i'm going to have my first term times the derivative of the second, which is just going to be negative 0 .06 times the original function e to the negative 0 .06t.

00:40 And then for the second part of my product rule i'm going to have 20 times the derivative of t, which in this case is going to be 1, times the second term e to the negative 0 .06t.

00:52 So i can go ahead and clean this up a little bit by factoring out the e term and the 20.

00:57 And i'll end up with something that looks like this.

00:59 I want to go ahead and maximize this, so i'm going to set this equal to 0 and solve.

01:03 Well i know that unless time goes to negative infinity, this term here is never going to be equal to 0.

01:10 Or sorry, if time goes to infinity it will.

01:11 But we're not talking about infinite times here.

01:13 So i really just want to solve this part here.

01:16 So where negative 0 .06t plus 1 equals 0, t there is going to be 1 divided by 0 .06.

01:27 And that's going to be 16 .67 if i plug that into my calculator.

01:32 And t is in minutes here.

01:34 So that's where i'm going to have either a max or a min.

01:37 I want to make sure that that is in fact a maximum.

01:40 And so to do that i'm going to opt to do a quick first derivative test.

01:45 So i have from 0 to 16 .67, that's my interval of t.

01:50 And then from 16 .67 to time forever, i want to figure out what the value of c prime is.

01:57 So i could actually go ahead and just plug in 0 into c prime of t here.

02:02 That would be fine.

02:02 I'm just choosing 0 because time has to be a positive value.

02:05 But if i plug in 0 to here, i'm going to get 20 times 1 times 1.

02:10 So that's going to be a positive value.

02:12 If i go ahead and plug in say 50 ,000, well i know this term here is going to always be positive because it's e to some power...

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