Section 5.6: HISTORICAL AND EXPONENTIAL DEPRECIATION OBJECTIVES Write, interpret, and graph an exponential depreciation equation. Manipulate the exponential depreciation equation in order to determine time, original price, and depreciated value. Key Terms dollar value historical data historical depreciation exponential decay exponential depreciation Chapter 5: Automobile Ownership 1
Car Prices Over Time: What Happens? Recall: Most cars will not be worth their purchase prices as they get older. Most cars depreciate or lose value over time. In section 5.5, the car lost the same amount of dollar value each year and can be modeled by a straight line depreciation equation. Another approach to determine the way a car loses value is to look at prices from the past using historical data. The devaluation of a car when using this type of data is called historical depreciation. Kelley Blue Book is essentially a database of the prices of used cars. Exponential Depreciation What happens to the value of the car as time increases as shown in the scatterplot? The value of the car seems to have a greater decrease at the beginning of the car s lifetime and less as each year passes. The depreciation is not constant from year to year. This scatterplot shows an exponential decay function. In terms of auto devaluation, this model is exponential depreciation. The value of the car decreases by the same percentage each year. Chapter 5: Automobile Ownership 2
Exponential Depreciation General Form of the Exponential Depreciation Equation P = Initial Value of the Car r = percent of depreciation (expressed as a decimal) t = time in years y = value of the car after t years *Note: The extent to which the exponential depreciation model fits the historical data varies from situation to situation. Example 1 Using exponential regression, determine an exponential depreciation equation that models the data in the table. Round coefficients to the nearest hundredth. Chapter 5: Automobile Ownership 3
Example 2 What is the depreciation percentage rate for the 10 years of car prices as modeled by the equation? Recall: 1 Example 3 After entering a set of automobile value data into a graphing calculator, the following exponential regression equation information is given in the table. Determine the depreciation percentage rate. Round to the nearest tenth of a percent. 32,567.98722 0.875378566 Chapter 5: Automobile Ownership 4
Example 4 After graduating college, you financed a new car for $21,560. After thorough research, you learned that the value of the car will exponentially depreciate at a rate of 10.25% per year. (a) Write the exponential depreciation equation for this car. (b) Determine the value of the car after 80 months. Example 5 A car exponentially depreciates at a rate of 6% per year. Your neighbor purchased a 5 year old car for $18,000. What was the original price of the car when it was new? Round to the nearest hundred dollars. Chapter 5: Automobile Ownership 5
Example 6 Joey purchased a four year old car for $16,400 so he could drive to Westchester to visit Monica and Chandler. When the car was new, it sold for $23,000. Find the depreciation rate to the nearest tenth of a percent. Example 7 A car originally sold for $26,600. It depreciates exponentially at a rate of 5.5% per year. When purchasing the car, you put $6,000 down and pay $400 per month to pay off the balance. (a) Write an expense function to model your situation. (b) Write an exponential depreciation function to model your situation. Chapter 5: Automobile Ownership 6
Example 7 Continued (c) Using the graphing calculator, after how many years will the car value equal the amount you paid to date for the car? Exponential Depreciation Recall: General Form of the Exponential Depreciation Equation When you need to solve for the variable, t, in the equation, the length of time, t, can be determined by using the formula Since the variable is located in the exponent, the formula must contain a logarithm! P = Initial Value of the Car r = percent of depreciation (expressed as a decimal) t = time in years y = value of the car after t years Chapter 5: Automobile Ownership 7
Example 8 Leah and Brandon bought a used car valued at $20,000. When this car was new, it sold for $24,000. If the car depreciates exponentially at a rate of 8% per year, approximately how old is the car? Chapter 5: Automobile Ownership 8