Video Transcript
The number of people infected with
a virus is increasing at a rate of 17 percent a year. In the last year, 12,500 people
were infected with the virus. Write an equation that can be used
to calculate 𝑃, the number of people expected to be infected with the virus in the
next 𝑚 months. Then, the next part of the question
says, “How many people would be expected to catch the virus in the next seven
months?” Give your answer to the nearest 100
people.
So, in order to solve this problem,
what we’re gonna use is something here, which is the general form for an exponential
equation. And what that tells us is that the
function of 𝑥 is equal to 𝐴, where 𝐴 is the initial value, multiplied by 𝑏,
where 𝑏 tells us something about the rate. And 𝑏 is always positive and not
equal to one. And 𝑥 is the independent variable,
which is usually the number of time periods.
So the first thing we need to do is
work out what each of these values is going to be in our problem. But our 𝐴, so our initial value,
is gonna be 12,500 people because we know this was the starting point from the
question. And then our 𝑏 is gonna be
1.17. And the reason we know this is
because we’re told that the virus infections are increasing at a rate of 17 percent
a year. Well, if we think about what 100
percent means, well, 100 percent means 100 out of 100. So it’s just equal to one.
Well then, if we were to increase
this by 17 percent, because it says that the infections are increasing at this rate,
then what we’d get is 117 percent. Well, 117 percent means 117 out of
100, which is the same as 1.17, which is our decimal multiplier. Well then, finally, our 𝑥 is gonna
be equal to 𝑚 over 12. And the reason it’s that is because
we’re told the rate is 17 percent per year.
However, what we’re asked to do is
find an equation that can be used to calculate 𝑃, the number of people expected to
be infected with the virus in the next 𝑚 months. So we want it in terms of
months. So, therefore, if we do 𝑚 divided
by 12, that’s because there are 12 months within a year. So, therefore, if we put this all
together, we’re gonna get the equation 𝑃 equals 12,500 multiplied by 1.17 to the
power of 𝑚 over 12. Okay great, so that’s the first
part of our question answered.
Now let’s move on to the second
part. And in the second part, what we
want to know is how many people would be expected to catch the virus in the next
seven months. Well, to solve this problem, all we
need to do is use the equation that we formed in the first part. So we know that we want to
calculate the number of people who are expected to catch the virus in the next seven
months. So, therefore, our 𝑚 is going to
be equal to seven because 𝑚 represents the number of months. So we’re gonna substitute this
in.
So, when we’ve done that, we’re
gonna get 𝑃 is equal to 12,500 multiplied by 1.17 to the power of seven over
12. And when we calculate that, we get
13,698.8811 et cetera. Well, have we finished there, we’ve
worked out the answer? Well, no, we’re not quite
finished. And that’s because the question
asks us to give the answer to the nearest 100 people. So, therefore, we can say that to
the nearest 100 people, we would expect 13,700 to catch the virus in the next seven
months.
