Lance buys a new car for $20,000. The value of the car depreciates by 15% each year. If [tex]f(x)[/tex] represents the value of the car after [tex]x[/tex] years, which function represents the car's value?
A. [tex]f(x)=20,000(0.85)^x[/tex]
B. [tex]f(x)=20,000(0.15)^x[/tex]
C. [tex]f(x)=20,000(1.15)^x[/tex]
D. [tex]f(x)=20,000(1.85)^x[/tex]
Answer (2)
The problem describes an exponential decay situation where a car's value depreciates by 15% each year.
The general form of the exponential decay function is f ( x ) = A ( 1 − r ) x , where A is the initial value and r is the depreciation rate.
Substituting the given values, A = 20000 and r = 0.15 , into the formula.
The function representing the car's value after x years is f ( x ) = 20000 ( 0.85 ) x .
Explanation
Understanding the Problem We are given that Terence buys a new car for $20,000. The value of the car depreciates by 15% each year. We need to find a function f(x) that represents the car's value after x years.
General Form of Exponential Decay The general form of an exponential decay function is given by: f ( x ) = A ( 1 − r ) x where:
A is the initial value
r is the depreciation rate (as a decimal)
x is the number of years
Identifying the Values In this case, the initial value of the car is A = 20000 , and the depreciation rate is r = 15% = 0.15 .
Substituting the Values Substitute the given values into the formula: f ( x ) = 20000 ( 1 − 0.15 ) x
Simplifying the Expression Simplify the expression inside the parentheses: 1 − 0.15 = 0.85
Final Function Therefore, the function that represents the car's value after x years is: f ( x ) = 20000 ( 0.85 ) x
Examples
Understanding depreciation is crucial in personal finance. For instance, when buying a car, its value decreases over time due to wear and tear and market factors. The formula we derived helps predict the car's future value, aiding in decisions about when to sell or trade it in. Similarly, businesses use depreciation to account for the declining value of assets like machinery, affecting their tax liabilities and financial planning. This concept extends to other assets like electronics and real estate, where understanding depreciation helps in making informed investment and financial decisions.
The function that models the value of Lance's car after a certain number of years, considering a 15% annual depreciation, is f ( x ) = 20 , 000 ( 0.85 ) x . This reflects a decrease of 15% each year from the initial value of $20,000. The correct answer is option A.
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