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The value of a new car is $30,000, and it loses 12% of its value each year. What is the exponential function modeling the car's value over time?

A) \( V(t) = 30000 \times (0.88)^t \)
B) \( V(t) = 30000 \times (1.12)^t \)
C) \( V(t) = 30000 \times (0.12)^t \)
D) \( V(t) = 30000 \times (0.98)^t \)

Answer :

Final answer:

The correct exponential function modeling the car's value over time, with a 12% annual depreciation, is V(t) = 30000 x (0.88)^t.

Explanation:

The question revolves around finding an exponential function that accurately represents the depreciation of a car's value over time. Given that a new car is initially valued at $30,000 and depreciates by 12% each year, it follows a predictable pattern of value reduction. Specifically, with each passing year, the car retains 88% of its previous year's value, as depicted by the expression (0.88)^t in the model V(t) = 30000 x (0.88)^t, where t represents the number of years elapsed.

This exponential decay model effectively encapsulates the dynamics of the car's depreciation, offering a clear understanding of how its value diminishes over time. As t increases, the exponentiation of 0.88 to the power of t accounts for the cumulative effect of annual depreciation on the car's worth. Thus, by plugging in different values for t, one can precisely calculate the car's value at any given point in time, facilitating informed decision-making regarding its purchase, resale, or overall financial planning.

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