The current value of a new car is $24,000. The car will depreciate 12% per year over the next 5 years. Which exponential equation models this situation?
A. f(x)=24,000 * 0.12^x
B. f(x)=24,000 * 0.88^x
C. f(x)= 24,000 * 5^x
D. f(x)= 24,000 * 1.12^x
The Correct Answer and Explanation is :
The correct equation is B. f(x) = 24,000 * 0.88^x.
Explanation:
Depreciation refers to the loss in value of an asset over time. In this case, the car loses 12% of its value each year. To model this depreciation using an exponential function, we start with the general form of an exponential decay equation:
[
f(x) = P \cdot (1 – r)^x
]
Where:
- ( f(x) ) is the value of the car after ( x ) years,
- ( P ) is the initial value of the car,
- ( r ) is the rate of depreciation (in decimal form), and
- ( x ) is the number of years.
In this case:
- The initial value ( P = 24,000 ),
- The depreciation rate ( r = 12\% ) or 0.12 in decimal form,
- The decay factor ( 1 – r = 1 – 0.12 = 0.88 ), which represents retaining 88% of the car’s value each year.
Thus, the equation that models the car’s depreciation is:
[
f(x) = 24,000 \cdot 0.88^x
]
This means that after each year, the car retains 88% of its value from the previous year.
Why not the other options?
- A. f(x) = 24,000 * 0.12^x: This suggests that the car would retain only 12% of its value each year, which is incorrect as the car loses 12%, not keeps only 12%.
- C. f(x) = 24,000 * 5^x: This suggests exponential growth rather than decay, which is not applicable in this depreciation scenario.
- D. f(x) = 24,000 * 1.12^x: This suggests that the car’s value increases by 12% per year, which is the opposite of what’s happening.