A house was valued at 105,000
in the year 1994.
The value appreciated to 165,000 by the year 2007.
We're first asked what was the annual growth rate
between 1994 and 2007.
Then we're asked to enter the correct answer
in part A as a percentage,
so we'll express the growth rate as a decimal in part A,
and as a percentage in part B,
and then for part C, if we assume that the house value
continues to grow by the same percentage,
what will the value equal in the year 2010?
And we're asked to round the value
to the nearest thousand dollars.
This problem is an example of an application
involving exponential growth.
In our textbook,
we're given the following equation for exponential growth.
P sub-n equals the quantity one plus r,
raised to the power of n, times P sub-zero,
but we can also write this as P sub-zero,
times the quantity one plus r, raised to the power of n.
Where P sub-zero is the starting amount,
r is the growth rate as a decimal,
n is time, and P sub-n is the amount after n time periods.
However, a lot of other sources
may express an equivalent equation in this form here,
f of t equals a times b raised to the power of t.
It's important to recognize
that these two equations are equivalent,
they're just using different variables.
Where a is the starting amount,
which is the same as P sub-zero.
b, the base, is equal to one plus r,
where r is the growth rate as a decimal,
and t is time, which is the same as n in this equation here.
Let's head up and solve this problem
using this form of the equation for exponential growth.
The first value is given for the year 1994,
which is the starting year or base year,
so if n equals zero, we need to recognize that
this represents the year 1994.
This also tells us that P sub-zero
is equal to $105,000,
because this will be the starting amount,
or the starting value of the home.
And then we know the value is 165,000 in the year 2007.
So we need to figure out what n would be for the year 2007.
To do that, we take the desired year of 2007,
subtract the base year of 1994,
and this difference is equal to 13,
and now we know that P sub-13 is the value in 2007,
which equals $165,000.
This is all the information we need
in order to set up an equation so we can then solve for r.
Let's go ahead and use this form of the equation here,
where we have P sub-zero on the left, rather than the right.
To write our equation, we'll let P sub-n
be equal to P sub-13, so our equation's going to be
165,000 equals P sub-zero,
the starting amount or starting value of 105,000,
times the quantity one plus r, where r is the unknown,
and now for the value of n,
because we have 165,000 on the left,
we know that n is equal to 13.
So our goal here is to solve this equation for r.
So for the first step, let's isolate the quantity one plus r
raised to the power of 13.
So we'll divide both sides by 105,000.
On the right side, this quotient is equal to one.
Let's find this quotient on the calculator.
So we have 165,000
divided by 105,000.
Enter.
And we don't want to use a rounded decimal here,
we want the exact value.
So we'll convert this decimal to a fraction
by pressing Math, Enter, Enter.
And so we have 11/7.
So now we have the equation 11/7 equals,
we have the quantity one plus r, raised to the power of 13.
Now here to solve for r, you want to undo this power of 13.
To do this, we'll use the property of exponents square.
If we have a raised to the power of m,
raised to the power of n, we multiply the exponents.
This is equal to a raised to the power of m times n.
So to undo this exponent, we want the exponent to be one.
So we need to think, what do we multiply 13 by
to get a product of one?
It would be the reciprocal of 13,
so 13 as a fraction is 13 over one.
So 13 over one times the reciprocal of one over 13
gives a product of one,
which means we're going to raise both sides of the equation
to the reciprocal power of 13, which is 1/13.
So I'll raise the right side to the 1/13 power,
and we'll also raise the right side to the 1/13 power.
So now on the right side,
when we multiply these exponents, we get a product of one,
and the quantity one plus r to the first is just one plus r.
So we have one plus r equals,
on the left side we have 11/7 raised to the power of 1/13.
Let's continue on the next slide.
To solve for r,
we would subtract one on both sides of the equation,
so we'd have r equals
11/7 raised to the power of 1/13 minus one.
And now we'll go to the calculator
and get a decimal approximation for the value of r.
So in parentheses, we have the fraction 11/7.
We want to raise this to the power of 1/13,
so here's the exponent key,
and the exponent is one divided by 13,
press the right arrow to exit the exponent,
and then we have minus one, Enter.
And we are told to round to four decimal places,
so r is approximately 0.0354.
So going back to our first slide,
this is the answer to part A, 0.0354.
Part B, we're asked to convert the decimal to a percentage.
To convert a decimal into a percentage,
we multiply it by 100,
or move the decimal point to the right two places.
And therefore our percentage is 3.54%.
Now we're going to make a prediction
about the value of the home to the nearest thousand dollars
for the year 2010.
So the general equation for P sub-n
is P sub-n equals, again the initial or starting value,
which is 105,000, times the quantity one plus r
raised to the power of n,
so we have one plus 0.0354, raised to the power of n.
Of course we can find this sum here,
and write P sub-n as P sub-n equals 105,000
times 1.0354 raised to the power of n.
Now we need to figure out what n would be for the year 2010.
Well n would be equal to the desired year of 2010,
minus the base year.
Remember the base year was the starting year of 1994.
So minus 1994, and this difference is equal to 16,
which means to approximate the value in 2010,
we need to find P sub-16.
So P sub-16 is equal to 105,000 times 1.0354
raised to the power of 16.
Back to the calculator.
So we have 105,000
times 1.0354 in parentheses,
raised to the power of 16. Enter.
So rounding to the nearest cent,
we have approximately $183,197.64.
Normally we do round money to the nearest cent,
but in this case, going back to the directions,
they do say to the nearest thousand dollars.
So here's the thousands place value,
and because the digit in the hundreds place value's a one,
we round down to $183,000.
So our approximate value is $183,000
for the year 2010.
I hope you found this helpful.