Solve the problem. The attending physician in an emergency...

Solve the problem. The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration A0 of the drug in the bloodstream at the time of injection. However, the physician knows that after 3 hr. the drug concentration in the blood is 0.75 ?g/dL and after 4 hr. the concentration is 0.719 ?g/dL. The model A(t) = A0 e^-kt represents the drug concentration A(t) in ?g/dL in the bloodstream t hours after injection. The value of k is a constant related to the rate at which the drug is removed by the body. Select one: a. a. 0.75 = A0e^3k b. 0.719 = A0e^4k c. k = 0.618 b. a. 0.75 = A0e^-3k b. 0.719 = A0e^-4k c. k = 0.042 c. a. 0.75 = A0e^-3k b. 0.719 = A0e^-4k c. k = 0.618 d. a. 0.75 = A0e^3k b. 0.719 = A0e^4k c. k = 0.042

Transcript

00:01 All right.

00:01 This one, um, is an exponential function, right? and it starts to give you a little bit of information, right? um, and it gives you an equation, i believe, which we're going to write down here.

00:14 It's a of t equals a naught, a naught, a sub zero.

00:19 That's always your initial amount.

00:20 E to the negative k t power.

00:23 And it says the physician knows that after three hours, the drug concentration is .75.

00:31 So we know that that final amount is .75.

00:34 We don't know what we're starting with, and we don't know the rate at which it is decaying, but we know after three hours it's .75.

00:44 And then after four hours, it's .719.

00:50 So .719 equals a naught, e to the negative k times four.

00:57 So if you notice, we have two unknowns here.

00:59 We have an unknown for our initial amount and we have an unknown for our rate.

01:04 All right.

01:04 So they want you to be able to set up two separate equations, right? so when you look at your paper, um, that's the wrong one.

01:13 Let's see if i can find it.

01:14 There it is.

01:15 Um, you can eliminate the first one and the last one because the, the k isn't negative, right? so that puts us at, um, i don't know, bbc and cbc, whatever those are.

01:29 All right.

01:30 So those are our two things right now.

01:33 So then we just have to figure out what the k value is.

01:36 All right.

01:36 So the only way we can do that is we can set both of these equal to a naught.

01:41 All right.

01:42 So a sub zero is going to be .75 over e to the negative three k and a naught over here is going to be .719 over e to the negative four k.

01:57 And now since both of them are equal to the same thing, a naught, we can set them equal to each other.

02:03 So .75 e to the negative three k is going to equal .719 e to the negative four k.

02:12 Now what i did here is i, i cross multiplied.

02:18 All right.

02:18 Cause i have a proportion, right? so in cross multiplying, i'm going to get .75 times e to the negative four k equals .719 e to the negative three k.

02:31 So far so good.

02:34 All right.

02:35 I want to bring the e to the negative three k over.

02:38 So i'm going to divide by that.

02:40 All right.

02:42 When i divide by that, these guys will cancel.

02:45 Anything over itself is, will be one.

02:47 And this thing on the left here, i have common bases...

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