Doug bought a new car for $25,000. He estimates his car will depreciate, or lose value, at a rate of $20\%$ per year. The value of his car is modeled by the equation [tex]V=P(1-r)^{\prime}[/tex], where [tex]V[/tex] is the value of the car, [tex]P[/tex] is the price he paid, [tex]r[/tex] is the annual rate of depreciation, and [tex]t[/tex] is the number of years he has owned the car. According to the model, what will be the approximate value of his car after [tex]4 \frac{1}{2}[/tex] years?
A. $2,500
B. $9,159
C. $22,827
D. $23,791
Answer (2)
The approximate value of Doug's car after 4 2 1 years will be around $9,159 based on the depreciation rate of 20% per year.
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Identify the initial price P = $25 , 000 , depreciation rate r = 0.20 , and time t = 4.5 years.
Substitute these values into the depreciation formula: V = P ( 1 − r ) t = 25000 ( 1 − 0.20 ) 4.5 .
Simplify the expression: V = 25000 ( 0.8 ) 4.5 .
Calculate the approximate value: V ≈ $9 , 159 .
Explanation
Identify Given Information First, we need to identify the given information from the problem.
The initial price of the car, P , is $25 , 000 .
The annual depreciation rate, r , is 20% , which can be written as 0.20 .
The time, t , is 4 2 1 years, which is equal to 4.5 years. The value of the car, V , is modeled by the equation V = P ( 1 − r ) t .
Substitute Values into Equation Now, we substitute the given values into the equation:
V = 25000 ( 1 − 0.20 ) 4.5
Simplify Expression Next, we simplify the expression inside the parentheses:
1 − 0.20 = 0.80
So the equation becomes:
V = 25000 ( 0.8 ) 4.5
Calculate the Value of V Now, we calculate the value of V :
V = 25000 × ( 0.8 ) 4.5 ≈ 25000 × 0.366356 ≈ 9158.90
Rounding to the nearest dollar, we get $V \approx $9,159.
Final Answer Therefore, according to the model, the approximate value of Doug's car after 4 2 1 years will be $$9,159.
Examples
Understanding depreciation is crucial in personal finance. For instance, when buying a car, its value decreases over time. The formula V = P ( 1 − r ) t helps estimate this decrease, where V is the future value, P is the initial price, r is the depreciation rate, and t is the time in years. This calculation aids in making informed decisions about insurance, resale value, and overall financial planning, ensuring you're prepared for the long-term financial implications of owning the car. For example, knowing that a $25 , 000 car depreciating at 20% annually will be worth approximately $9 , 159 after 4.5 years helps in planning for its replacement or sale.
