Final answer:
To find the time until the car depreciates to $10,800, we use the exponential decay formula. After solving the equation, we find that it will take approximately 8 years for the car to depreciate to that value.
Step-by-step explanation:
The question asks how long it will take for a car, purchased for $20,700, to depreciate to $10,800 at an annual depreciation rate of 7.25%. This is a typical exponential decay problem, where we can use the formula for exponential decay to find the time, t:
V = P(1 - r)^t
Where:
- V is the future value of the car
- P is the initial value of the car
- r is the depreciation rate (as a decimal)
- t is the time in years
Plugging in the given values:
10800 = 20700(1 - 0.0725)^t
To find t, we need to solve for it using logarithms:
- Divide both sides by 20700:
(10800 / 20700) = (1 - 0.0725)^t - Calculate the left side and take the natural logarithm of both sides:
ln(10800 / 20700) = ln((1 - 0.0725)^t) - Apply the power rule of logarithms to move t out front:
t * ln(1 - 0.0725) = ln(10800 / 20700) - Solve for t:
t = ln(10800 / 20700) / ln(1 - 0.0725) - Calculate the value of t using a calculator.
When you do the math, you will find that t is approximately 8 years (rounding to the nearest year).