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Hey guys, Skylar here. I'm at Jim's to Junk Auto Sales with Jill.
Jill's in the market for a set of wheels. Her first car. Now,
she sees herself in a sports car, but I think one of these beauties
is a little more in line with her budget. I a few years, Jill hopes
to trade in her first car for her dream car. When making her selection,
she'll have to keep resale value in mind and understand how cars
depreciate. What will help is a clear understanding of exponential decay.
So while Jill goes and kicks a few tires, let's take a drive through decay and
get another problem solved.
Maybe you need to keep looking.
To begin you have to
understand that the value of Jill's car will decrease over time. I'll use
this car to explain. Right now the car is worth $10,000.
$10,000 is 100% of the car's value.
Assume the value decreases, or decays, at a rate of
12%, or 12 hundredths, per year. This
is the decay rate. Over the next year, the car's value will
depreciate, or decay, by 12% of $10,000,
which is $1,200. Ouch.
Let's find its value after one year. We'll take 100%
of the original value and subtract 12% of the original
value. So $10,000 times 100%,
or one, minus $10,000 times
12%, or 12 hundredths, is equal to
10,000 times one minus .12,
or 10,000 times .88, which equals
$8,800, the value one year from now.
12%, or 12 hundredths, is our decay rate.
But the decay factor is one minus
.12, which equals .88, or
88%. The decay factor is the amount of value
the car maintains from year to year. It's always between
zero and one. Now, let's create
a table and graph to estimate the value of the car two
years from now. To find the value at the end of the second year, we take
the estimated value of the car after the first year, and multiply it
by the decay factor of .88, which
is 7,744. If we continue
the process of multiplying our new value by the decay factor,
we predict it will be approximate $6,815
by the end of the third year. I'm gonna hit the breaks
for a sec. Look at this. During the first year, the car
value was predicted to depreciate by $1,200.
But during the second year the car's value is predicted to
depreciate by less than $1,200.
The depreciation is not the same from year to year.
Let's say Jill wants to find out what the value of the car will be in ten years.
She could continue to multiply by .88 for each of
the ten years, but as you may have guessed, there is another way. Lets
go back to the table. We started with $10,000
and multiplied by our decay factor of .88.
Then, to find the estimated value of the car at the end of
the second year, we multiplied our value at the end of year one
by .88. Remember, 10,00 times
.88 is equal to 8,800, so to help us find
a pattern, we're gonna replace 8,800
with 10,000 times .88.
For the third year, we multiply the second year's value by .88.
Lets write 7,744 as
10,000 times .88 times
.88. Look at the calculations through the third year.
The pattern's right there. We repeatedly multiplied by the decay factor.
We can go a step further by using exponents. Exponents
help us calculate the predicted value for any year, which is represented
by n. We simply take 10,000
times .88, raised to the nth power.
If we want to predict what the car's value would be in ten years,
we take $10,000 times .88
raised to the tenth power, which is approximately
$2,785. So if Jill bought this
car today for $10,000, she could expect to resell or
trade it in for about $2,785
ten years from now. Now, we're talking cars today,
but we can calculate decay for any situation if we can identify
just two important values. The value of the
dependent variable when n equals zero, which is a, and
the decay factor, which is one minus r. Why don't we
try identifying these two special values in our equation?
Remember, a is the value of the dependent variable when n equals zero.
Notice in our equation this value, a, is 10,000.
The second important value is the decay factor.
In our example, the decay factor is .88.
This is one minus the decay rate, r.
Lets generalize our result. Instead of using v for value
and n for number of years, we can replace these variables with
y and x. So any exponential decay equation
can be represented by y equals a times the quantity
one minus r to the x power. There,
you're ready to roll. Here's something to remember.
The greatest decay always occurs right away, then lessens
over time. In the first year, the car's value will decrease by a whopping
$1,200. But in the tenth year, we're
predicting it will only decrease by about $380.
Turns out, Jill is a pretty shrewd negotiator. As she closes the
deal, keep in mind there are always a lot of factors influencing actual
decay. You should be ready to apply your understanding of exponential decay in similar
situations. Problem solved.