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So polluted water is passed through a series of filters.
Each filter removes 90% of the remaining impurities from the water.
so if we have 10 million particles of pollutant per gallon originally, how many
filters would The, what the water need to be passed through to reduce the pollutant
of 500 particles per gallon, so here we're going to set up an equation where
our poulation, if you will, is representing the pollutant per gallon.
An n instead of representing years here is going to represent the number of
filters that we apply To our to, that we apply to our water.
So our model then is going to say, well what's our P zero?
P zero is the initial amount. Our initial amount is 10 million
particles. And how is it changing.
Well it's not growing, right. It's decreasing, so we're going to have 1
minus, right, because if not we're going to have a positive growth rate.
We're going to have a negative growth rate.
We are decreasing by 90%, not each year but for each filter.
And so our equation is going to look like this.
Or more simply 10 million times 1 minus .9 is .10 to the n.
Now to answer our question, we want to know when the pollutant will be 500 so
we're going to set our equation equal to 500.
So now, in order to start solving this for n, the first thing we need to do is
get that exponential part by itself. So we're going to divide both sides of
the equation by 10,000,000. So on the right I've got 0.10 to the n.
Now if we can avoid decimals great but if not, we've got 500 divided by 10, 1 2 3 4
5 6, 10 million. Is 0.00005.
Okay? So that's 0.00005 equals 0.10 to the end.
Now to get, in order to be able to solve for that exponent we are now going to
need our logorithm, so we're going to apply the log function to both sides.
And we can utilize our cool x property of logarithms which says that if we got the
log of something raised to a power, that's the same as, the exponent times
the log of the base of that exponent. Okay, and now we can solve for we can
solve for N. Now we can either evaluate each of these
as decimals now or later. Let's go ahead and do it now.
So we already have our .005 here, let's take the log of that, is negative 4.301.
So what's here negative 4.301 equals n times, now lets see 0.10 log is negative
1, okay so we got negative 1 as the log of 0.10 so now we can divide by negative
one And we end up with, we end up with n equals 4.301, because our negatives
cancel here filters. Now of course we can't really have 0.4
fill 0.3 filters, so if wanted to reduce the pollutant, below a 500 particles per
gallon Or really need five filters.