Exponential Depreciation
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Transcript Exponential Depreciation
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