In this final section of this chapter we need to look at
some applications of exponential and logarithm functions.
Compound Interest
This first application is compounding interest and there are
actually two separate formulas that we’ll be looking at here. Let’s first get those out of the way.
If we were to put P
dollars into an account that earns interest at a rate of r (written as a decimal) for t
years (yes, it must be years) then,
- if
interest is compounded m times
per year we will have
dollars
after t years.
- if
interest is compounded continuously then we will have
dollars
after t years.
Let’s take a look at a couple of examples.
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Example 1 We
are going to invest $100,000 in an account that earns interest at a rate of
7.5% for 54 months. Determine how much
money will be in the account if,
(a) interest
is compounded quarterly. [Solution]
(b) interest
is compounded monthly. [Solution]
(c) interest
is compounded continuously. [Solution]
Solution
Before getting into each part let’s identify the
quantities that we will need in all the parts and won’t change.
Remember that interest rates must be decimals for these
computations and t must be in
years! Now, let’s work the problems.
(a) Interest is compounded
quarterly.
In this part the interest is compounded quarterly and that
means it is compounded 4 times a year.
After 54 months we then have,
Notice the amount of decimal places used here. We didn’t do any rounding until the very
last step. It is important to not do
too much rounding in intermediate steps with these problems.
[Return to Problems]
(b) Interest is compounded
monthly.
Here we are compounding monthly and so that means we are
compounding 12 times a year. Here is
how much we’ll have after 54 months.
So, compounding more times per year will yield more money.
[Return to Problems]
(c) Interest is compounded
continuously.
Finally, if we compound continuously then after 54 months
we will have,
[Return to Problems]
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Now, as pointed out in the first part of this example it is
important to not round too much before the final answer. Let’s go back and work the first part again
and this time let’s round to three decimal places at each step.
This answer is off from the correct answer by $593.31 and
that’s a fairly large difference. So,
how many decimal places should we keep in these? Well, unfortunately the answer is that it
depends. The larger the initial amount
the more decimal places we will need to keep around. As a general rule of thumb, set your
calculator to the maximum number of decimal places it can handle and take all
of them until the final answer and then round at that point.
Let’s now look at a different kind of example with
compounding interest.
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Example 2 We
are going to put $2500 into an account that earns interest at a rate of
12%. If we want to have $4000 in the
account when we close it how long should we keep the money in the account if,
(a) we
compound interest continuously. [Solution]
(b) we
compound interest 6 times a year. [Solution]
Solution
Again, let’s identify the quantities that won’t change
with each part.
Notice that this time we’ve been given A and are asking to find t.
This means that we are going to have to solve an exponential equation
to get at the answer.
(a) Compound interest
continuously.
Let’s first set up the equation that we’ll need to solve.
Now, we saw how to solve these kinds of equations a couple
of sections ago. In that section we saw that we need to get
the exponential on one side by itself with a coefficient of 1 and then take
the natural logarithm of both sides.
Let’s do that.
We need to keep the amount in the account for 3.917 years
to get $4000.
[Return to Problems]
(b) Ccompound interest 6
times a year.
Again, let’s first set up the equation that we need to
solve.
We will solve this the same way that we solved the
previous part. The work will be a
little messier, but for the most part it will be the same.
In this case we need to keep the amount slightly longer to
reach $4000.
[Return to Problems]
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Exponential Growth
and Decay
There are many quantities out there in the world that are
governed (at least for a short time period) by the equation,
where is positive and is the amount initially
present at and k
is a non-zero constant. If k is positive then the equation will
grow without bound and is called the exponential
growth equation. Likewise, if k is negative the equation will die down
to zero and is called the exponential
decay equation.
Short term population growth is often modeled by the
exponential growth equation and the decay of a radioactive element is governed
the exponential decay equation.
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Example 3 The
growth of a colony of bacteria is given by the equation,
If there are initially 500 bacteria present and t is given in hours determine each of
the following.
(a) How
many bacteria are there after a half of a day? [Solution]
(b) How
long will it take before there are 10000 bacteria in the colony? [Solution]
Solution
Here is the equation for this starting amount of bacteria.
(a) How many bacteria are
there after a half of a day?
In this case if we want the number of bacteria after half
of a day we will need to use since t
is in hours. So, to get the answer to
this part we just need to plug t
into the equation.
So, since a fractional population doesn’t make much sense we’ll
say that after half of a day there are 5190 of the bacteria present.
[Return to Problems]
(b) How long will it take
before there are 10000 bacteria in the colony?
Do NOT make the mistake of assuming that it will be approximately
1 day for this answer based on the answer to the previous part. With exponential growth things just don’t
work that way as we’ll see. In order
to answer this part we will need to solve the following exponential equation.
Let’s do that.
So, it only takes approximately 15.4 hours to reach 10000
bacteria and NOT 24 hours if we just double the time from the first
part. In other words, be careful!
[Return to Problems]
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Example 4 Carbon
14 dating works by measuring the amount of Carbon 14 (a radioactive element)
that is in a fossil. All living things
have a constant level of Carbon 14 in them and once they die it starts to
decay according to the formula,
where t is in
years and is the amount of Carbon 14 present at death
and for this example let’s assume that there will be 100 milligrams present
at death.
(a) How
much Carbon 14 will there be after 1000 years? [Solution]
(b) How
long will it take for half of the Carbon 14 to decay? [Solution]
Solution
(a) How much Carbon 14
will there be after 1000 years?
In this case all we need to do is plug in t=1000
into the equation.
So, it looks like we will have around 88.338 milligrams
left after 1000 years.
[Return to Problems]
(b) How long will it take
for half of the Carbon 14 to decay?
So, we want to know how long it will take until there is
50 milligrams of the Carbon 14 left.
That means we will have to solve the following equation,
Here is that work.
So, it looks like it will take about 5589.897 years for
half of the Carbon 14 to decay. This
number is called the half-life of
Carbon 14.
[Return to Problems]
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We’ve now looked at a couple of applications of exponential
equations and we should now look at a quick application of a logarithm.
Earthquake Intensity
The Richter scale
is commonly used to measure the intensity of an earthquake. There are many different ways of computing
this based on a variety of different quantities. We are going to take a quick look at the
formula that uses the energy released during an earthquake.
If E is the energy
released, measured in joules, during an earthquake then the magnitude of the
earthquake is given by,
where joules.
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Example 5 If
joules of energy is released during an
earthquake what was the magnitude of the earthquake?
Solution
There really isn’t much to do here other than to plug into
the formula.
So, it looks like we’ll have a magnitude of about 7.
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Example 6 How
much energy will be released in an earthquake with a magnitude of 5.9?
Solution
In this case we will need to solve the following equation.
We saw how solve these kinds of equations in the previous section.
First we need the logarithm on one side by itself with a coefficient
of one. Once we have it in that form
we convert to exponential form and solve.
So, it looks like there would be a release of joules of energy in an earthquake with a
magnitude of 5.9.
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