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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 $V=P(1-r)^t$, where $V$ is the value of the car, $P$ is the price he paid, $r$ is the annual rate of depreciation, and $t$ is the number of years he has owned the car. According to the model, what will be the approximate value of his car after $4 \frac{1}{2}$ years?


A$\$2,500$
B$\$9,159$
C$\$22,827$
D$\$23,791$

Verified Answer :

  • The initial price of the car is \(P = \$25,000\) and the annual depreciation rate is \(r = 0.20\).
  • Using the formula \(V = P(1-r)^t\), substitute \(P=25000\), \(r=0.20\), and \(t=4.5\).
  • Calculate the resulting value: \(V = 25000 \times (0.8)^{4.5} \approx 9158.93\).
  • The value of the car after \(4 \frac{1}{2}\) years is approximately \(\boxed{\$9,159}\).

Explanation

  1. Problem Analysis and Data Identification

We are given an initial car price \(P = \$25,000\) and an annual depreciation rate \(r = 20\% = 0.20\). The value of the car after \(t\) years is modeled by the exponential decay formula \(V = P(1-r)^t\). We need to determine the value of the car after \(t = 4.5\) years (since \(4 \frac{1}{2} = 4.5\)).

  1. Calculation of Car Value

The formula for the car's value is:
\[V = 25000 \times (1 - 0.20)^{4.5}\]
Substituting the known values, the expression simplifies to:
\[V = 25000 \times (0.8)^{4.5}\]
We now calculate the value of \((0.8)^{4.5}\) and then multiply by \(25,000\). Using calculation, we find:
\[(0.8)^{4.5} \approx 0.366357\]
\[V \approx 25000 \times 0.366357 = 9158.934...\]
Rounding to the nearest dollar, the value is approximately \(\$9,159\).

  1. Conclusion

By comparing our calculated value to the given choices, we see that the value matches the option \(\$9,159\). Thus, the approximate value of the car after \(4.5\) years is \(\boxed{\$9,159}\).

Examples

Understanding exponential decay models is vital for managing finances and assets. For instance, when buying a new vehicle, machinery, or even technology, you can use similar depreciation models to forecast how much the item will be worth in the future. This helps individuals and businesses make informed decisions about when to resell items or how to plan for replacements over time. Applying these models allows for realistic financial planning and budgeting.

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