Exponential Depreciation
Section 5-6
Exponential Depreciation

In the previous section, we learned about
straight line depreciation


Cars lose the same value each year
If this is not the case, how could we
determine the value of a car after a
certain number of years

Look at historical data
Exponential Depreciation

Chevrolet Corvette 2-door Coupe in good
condition
Age
1
Value ($)
24,230
Age
6
Value ($)
15,245
2
3
4
22,355
20,645
18,070
7
8
9
14,075
13,100
12,325
5
16,256
10
11,525
Exponential Depreciation
Y-Values
26,000
24,000
22,000
20,000
18,000
16,000
14,000
12,000
10,000
0
2
4
6
8
10
12
Exponential Depreciation

The actual graph of this depreciation
appears to be curved


It is not a straight line
The car value seems to drop more at the
beginning of the car’s lifetime and less as
each year passes
Exponential Depreciation

This is known as exponential decay

The value of the car decreases by the same
percentage each year

Model is known as Exponential
Depreciation
Exponential Depreciation

Exponential Depreciation
Y = A (1 − r)x
A = starting value of car
r = percent of depreciation
x = time in years
Y = value of the car after x years
Exponential Depreciation

You buy a car that originally sells for
$25,000. It exponentially depreciates at a
rate of 4 ¼ % per year.

Write an exponential depreciation
equation for this car.
Y = 25,000 (1 − .0425)x
Exponential Depreciation


Tanya’s new car sold for $23,856. Her
online research indicates that the car will
3
depreciate exponentially at a rate of 6 %
8
per year.
Write an exponential depreciation
equation for this car.
Y = 23,856 (1 − .06375)x
Exponential Depreciation
You buy a used car for $12,500. The car
depreciates exponentially at a rate of 5 ¼%
per year. How much will the car be worth
after 5 years?
Y = 12,500(1 − .0525)𝑥
Y = 12,500(1 − .0525)5
Y = $9,545.66
Exponential Depreciation
Sharon purchased a used car for $24,600.
The car depreciates exponentially by 8%
per year. How much will the car be worth
after 5 years?
Y = 24,600(1 − .08)𝑥
Y = 24,600(1 − .08)5
Y = $16,213.41
Exponential Depreciation

Page 257, 2 - 6
Exponential Depreciation
Section 5-6
Exponential Depreciation

Exponential Depreciation
Y = A (1 − r)x
A = starting value of car
r = percent of depreciation
x = time in years
Y = value of the car after x years
Exponential Depreciation
A car exponentially depreciates at a rate of
6% per year. Beth purchased a 5-year old
car for $18,000. What was the original
price of the car when it was new?
𝑥
Y = 𝐴(1 − 𝑟)
18,000 = 𝐴(1 − .06)5
18,000 = A (0.94)5
A=
18,000
(0.94)5
A = $24,526.37
Exponential Depreciation
A car exponentially depreciates at a rate of
9% per year. Beth purchased a 7-year old
car for $14,500. What was the original
price of the car when it was new?
𝑥
Y = 𝐴(1 − 𝑟)
14,500 = 𝐴(1 − .09)7
14,500 = A (0.91)7
A=
14,500
(0.91)7
A = $28,059.39
Exponential Depreciation
Yesterday, we:
1) Found the equation for exponential
depreciation
2) Calculated the value of a car after “x”
number of years
 Today, we are going to find the rate of
depreciation of a car

Exponential Depreciation
You buy a 4-year old car for $16,400.
When the car was new, it sold for $23,000.
Find the depreciation rate.
Y = 𝐴(1 − 𝑟)𝑥
16,400 = 23,000(1 − 𝑟)4
𝟏𝟔, 𝟒𝟎𝟎
= (𝟏 − 𝒓)𝟒
𝟐𝟑, 𝟎𝟎𝟎
Exponential Depreciation
1
𝟏𝟔, 𝟒𝟎𝟎 1
( 𝟐𝟑, 𝟎𝟎𝟎 )4 =((𝟏 − 𝒓)𝟒 )4
16,400 1
(
)4
23,000
=1-r
16,400 1
𝑟 =1−(
)4 = 8.1%
23,000
Exponential Depreciation
Steps to finding rate
1) Substitute variables into the equation
2) Divide both sides by starting price (A)
3) Raise both sides to the reciprocal of x
4) Re-write equation with r isolated
5) Plug into calculator
6) Convert to a percent

Exponential Depreciation
Brad purchased a 5-year old car for
$14,200. When the car was new, it sold for
$24,000. Find the depreciation rate.
Y = 𝐴(1 − 𝑟)𝑥
14,200 = 24,000(1 − 𝑟)5
𝟏𝟒, 𝟐𝟎𝟎
= (𝟏 − 𝒓)𝟓
𝟐𝟒, 𝟎𝟎𝟎
Exponential Depreciation
(
1
𝟏𝟒, 𝟐𝟎𝟎 1
𝟓 5
5
=
(𝟏
−
𝒓)
)
(
)
𝟐𝟒, 𝟎𝟎𝟎
14,200 1
(
)5
24,000
=1-r
14,200 1
𝑟 =1−(
)5 = 10 %
24,000
Exponential Depreciation
Worksheet
Exponential Depreciation
Exponential Depreciation
Exponential Depreciation