1. Exponential Depreciation
Section 5-6

2. 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

3. Exponential Depreciation
Chevrolet Corvette 2-door Coupe in good condition
Age
Value ($) 1
24,230 6
15,245 2
22,355 7
14,075 3
20,645 8
13,100 4
18,070 9
12,325 5
16,256 10
11,525

4. Exponential Depreciation

5. 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

6. 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

7. 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

8. 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

9. Exponential Depreciation
Tanya’s new car sold for $23,856. Her online research indicates that the car will depreciate exponentially at a rate of % per year.
Write an exponential depreciation equation for this car.
Y = 23,856 (1 − ) x

10. 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

11. 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

12. Exponential Depreciation
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13. Exponential Depreciation
Section 5-6

14. 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

15. 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? Y = 𝐴(1 −𝑟) 𝑥 18,000 = 𝐴(1 −.06) 5 18,000 = A (0.94) 5
A = 18,000 (0.94) 5
A = $24,526.37

16. 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? Y = 𝐴(1 −𝑟) 𝑥 14,500 = 𝐴(1 −.09) 7 14,500 = A (0.91) 7
A = 14,500 (0.91) 7
A = $28,059.39

17. Exponential Depreciation
Yesterday, we:
Found the equation for exponential depreciation
Calculated the value of a car after “x” number of years
Today, we are going to find the rate of depreciation of a car

18. 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 𝟐𝟑,𝟎𝟎𝟎 = (𝟏−𝒓) 𝟒

19. Exponential Depreciation
( ) 1 4
( ) 1 4
𝟏𝟔,𝟒𝟎𝟎 𝟐𝟑,𝟎𝟎𝟎 = (𝟏−𝒓) 𝟒
( 16,400 23,000 ) = 1 - r
𝑟=1−( 16,400 23,000 ) 1 4
= 8.1%

20. Exponential Depreciation
Steps to finding rate
Substitute variables into the equation
Divide both sides by starting price (A)
Raise both sides to the reciprocal of x
Re-write equation with r isolated
Plug into calculator
Convert to a percent

21. 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 𝟐𝟒,𝟎𝟎𝟎 = (𝟏−𝒓) 𝟓

22. Exponential Depreciation
( ) 1 5
( ) 1 5
𝟏𝟒,𝟐𝟎𝟎 𝟐𝟒,𝟎𝟎𝟎 = (𝟏−𝒓) 𝟓
( 14,200 24,000 ) = 1 - r
𝑟=1−( 14,200 24,000 ) 1 5
= 10 %

23. Exponential Depreciation
Worksheet

24. Exponential Depreciation

25. Exponential Depreciation

26. Exponential Depreciation