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# Exponential model word problem: medication dissolve

Video transcript

- [Voiceover] Carlos has
taken an initial dose of a prescription medication. The relationship between the time, between the elapsed time t, in hours, since he took the first dose
and the amount of medication, M of t, in milligrams, in
his bloodstream is modeled by the following function. Alright, in how many hours
will Carlos have 1 milligram of medication remaining
in his bloodstream? So M of what t is equal to, so
we need essentially to solve for M of t is equal to 1 milligram. Because M of t outputs, whatever
value it outputs is going to be in milligram. So let's just solve that. So M of t is, they give us a definition, it's model is an exponential
function, 20 times e to the negative 0.8 t is equal to one. So let's see, we can
divide both sides by 20 and so we will get e to the negative 0.8 t is equal to one over 20, one over 20. Which we could write as 0.05, 0.05. I have a feeling we're gonna
have to deal with decimals here regardless. And so how do we, how do we solve this? Well one way to think about
it, one way to think about it if we took, what happens
if we took the natural log of both sides? And just a remember, a
reminder, the natural log is the logarithm base e. So actually let me write
this, let me write this a little bit differently. So this is zero and that is 0.05. So I'm gonna take the natural
log of both sides, so ln, ln. So the natural log, this
says, what power do I have to raise e to, to get to
e to the negative 0.8 t? Well I've got to raise e
to the, this simplifies to negative 0.8 t. Once again natural log this thing... let me clarify ln of e
to the negative 0.8 t. This is equivalent to if
I were to write log base e of e to the negative 0.8 t. What power do I have to raise e to, to get to e to the negative 0.8 t? Well I have to raise it to
the negative 0.8 t power. So that's why the left-hand
side simplified to this and that's going to be
equal to the natural log, actually I'll just
leave it in those terms, the natural log of 0.05, the natural log of 0.05 all of that and now we
can divide both sides by negative 0.8 to solve for t. So let's do that. So we divide by negative 0.8, divide by negative 0.8 and so t is going to be equal
to all of this business. On the left-hand side now we just have a t and on the right-hand side
we have all of this business which I think a calculator
will be valuable for. So let me get a calculator out. Clear it out and let's start with 0.05. Let's take the natural log, that's that button right over
there, the natural log. We get that value and we want
to divide it by negative .8. So divided by, divided by .8 negative. So we're gonna divide by
.8 negative is equal to, let's see they wanted us to
round to the nearest hundredth so 3.74, so it'll take 3.74, seven four hours for his dosage to go down to one milligram
where it actually started at 20 milligrams. When t equals zero it's 20 after 3.74 hours he's down in his bloodstream to one milligram. I guess his body has metabolized
the rest of it in some way.