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Studying for a test? Prepare with these 6 lessons on Exponential growth & decay.

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# Constructing exponential models (old example)

Video transskription

Caesium-137 is a
radioactive tracer element used to study
upslope soil erosion and downstream sedimentation. It has a half-life of
approximately 30 days. So this half-life
of 30 days, this means that, if you were
to start with 2 kilograms of caesium-137, that
30 days later you're going to have 1
kilogram of caesium-137. The other kilogram has
decayed into other things. And if you waited
another 30 days, you would have half a
kilogram of caesium-137. Assume that the amount A in
becquerels of caesium-137 in a soil sample is given by
the exponential function A is equal to c times r to the t--
where t is the number of days since the release of caesium-137
into soil and c and r are unknown constants. So this bears some explaining. What is this becquerel business? So normally, if I were
to talk about the amount of some element, I'd probably
be thinking in terms of mass and I might talk in
terms of kilograms. But some people might
also be referring to the amount of this
radioactive substance in terms of the amount of
radioactivity it produces. And becquerels is the
international unit of radioactivity-- named
after Henri Becquerel, who codiscovered radioactivity
with Marie Curie. So you could consider this
the amount of caesium-137 that causes A becquerels
of radioactivity. But either way, we can just
think of it as a quantity. But it's really
the quantity that causes A becquerels of activity. So just to be
clear, the amount is given by the
exponential function A is equal to-- let me just
rewrite it-- A is equal to c times r to the t power--
where t is the number of days since the release of the
caesium-137 in the soil and c and r are unknown constants. Fair enough? So let's just be clear,
this is days since release. In addition, assume that we
know that the initial amount of caesium-137 released in
the soil is 8 becquerels. Solve for the unknown
constants c and r. So the initial in the soil. That's when t is equal to 0. When no days have passed. So we could say that the amount
at times 0-- well, that's going to be equal to c
times r to the 0 power, which is just going to be
equal to c times 1, which is equal to c. And they tell us what A of 0 is. They say A of 0 is 8 becquerels. So this is going
to be equal to 8. So our constant here, the c is
just going to be equal to 8. What is the value
of the constant? We could just write
8 right over there. So the value of the
constant c is 8. What is the value
of the constant r? Round to the nearest thousandth. So we're starting with 8. So A of 0 is 8. How much are we going
to have after 30 days? And the reason why
I'm picking 30 days is-- that is the
half-life of caesium-137. So A of 30, remember our
t is in-- let me just switch colors just for fun. Remember, t is in days. So A of 30. So after 30 days, I
am going to-- if I want to use this
formula right over here, if I wanted to use
the description of this exponential function,
we already know that c is 8. It's going to be 8 times
r to the 30th power, which is going to be equal to what? Well, if we started
with 8, 30 days later, we're going to
have half as much. We're going to
have 4 becquerels. And now, we can use
this to solve for r. So you have 8 times r to the
30th power is equal to 4. Divide both sides by 8. You get r to the 30th power
is equal to 4 over 8-- which is the same thing as 1/2. And then, we can raise both
sides to the 1/30th power. r to the 30th--
but then, you could think of the 30th root
of that or raising that to the 1/30th
power-- that's just going to give us r is equal to
one half to the 1/30th power. And that is something
that's very hard to compute in your head. So I suggest you use
a calculator for that. And they hint
because we're going to round to the
nearest thousandth. So let's get a
calculator right out. And so we're talking about
one half to the 1/30 power. So we get 0.9771599--
it keeps going. But they tell us to round to
the nearest thousandth, so 0.977 rounded to the
nearest thousandth. And then, they finally say-- how
many becquerels of caesium-137 remain in our sample 150 days
after its release in the soil? Use the rounded value of
r, and round this number to the nearest hundredth. So just to be clear, we
already know c and r. We know that the amount of
caesium-137 in becquerels-- as a function of
time in days-- is going to be equal to 8
times 0.977 to the t power-- where t is the number of
days that have passed. And they're essentially
saying, well, how much do we have
left after 150 days? So they want us to
calculate what is A of 150? Well, that's going to be 8
times 0.977 to the 150th power. And so clearly, we need
a calculator for this. So let's calculate that. So it's going to be
8 times-- and they tell us to use our rounded value
of r, not the exact value of r. So it's going to be 8 times
0.977 to the 150th power. And they want us to round
to the nearest hundredth. 0.24. 0.24 becquerels is kind
of the radioactivity level of the caesium-137
that we have left over. Now, one interesting
thing is they asked us to use the
rounded value of r. So we used the
rounded value of r. Because this right over
here is a multiple of 30, you could actually-- in not
too difficult of a way-- find out the exact
value that's left over. And actually, you don't even
need a calculator for it. I encourage you to
pause your video and try to think about that. Find the exact value. Well, instead of writing
0.977, let's write A of t as being equal to 8 times our r. This is an approximate
value for r. If we wanted to be
a little more exact, we can say that our r is one
half to the 1/30th power. And we're going to raise
that to the t power. Or we could say A of
t is equal to 8 times one half to the t/30 power. If we raise something
to an exponent and then raise that
to an exponent, we can take the product
of those exponents. So that's one half
to the t/30 power. Let me actually do
that in another color. Let me do that in yellow. So that's 8 times one
half to the t/30 power. And actually, I don't need this
parentheses right over here. This is another way
to describe A of t. So what is A of 150? So A of 150 is going to be
equal to 8 times one half to the 150/30. Well, that's just 5. One half to the fifth power. Well, what's one half
to the fifth power? That is 1 to the fifth over
2 to the fifth or 1/32. So this right over here is 1/32,
which is equal to 8 over 32, which is equal to
1 over 4, which is equal to 1 over 4 or 0.25. So using our
approximation for r, we got 0.24 when we
rounded to the 150th power. So that's using our
approximation a lot. We're taking 150 of these and
multiplying them together, but it's not too far off
of what the real value is. And they asked us to
use the rounded value. But if we'd used the precise
value, the actual value, 0.25 becquerels
would be left over.