Final answer:
Using an exponential decay model, the calculation shows that it will take approximately 8.1 years for the car to depreciate to half its value, which means the closest time frame in the options given is 10 years (d). The Rule of 70 was also used for confirmation and yielded a similar result.
Step-by-step explanation:
A new car with a value of $28,000 is depreciating at a rate of 0.71% per month. To determine how many years it will take for its value to be half of what it was, we can use an exponential decay model. Since the car depreciates by 0.71% per month, we can express this monthly rate as a decimal by dividing by 100, which gives us 0.0071. The depreciation can be represented by the formula:
V = P(1 - r)^t
Where V is the final value, P is the initial value, r is the depreciation rate per time period, and t is the number of time periods. We want to find out when the car's value will be half of its original value, so we set V as $14,000, which is half of $28,000, and P as $28,000. We are looking for t, the number of months it will take to reach this value. To solve for t, we can use the formula:
14,000 = 28,000(1 - 0.0071)^t
Dividing both sides by 28,000 we get:
0.5 = (1 - 0.0071)^t
Now we can use logarithms to solve for t. Taking natural logarithms of both sides gives us:
ln(0.5) = t * ln(1 - 0.0071)
Solving for t gives:
t = ln(0.5) / ln(1 - 0.0071)
Using a calculator, we find that t ≈ 97.22 months. Since we are looking for the number of years, we divide this by 12 months per year to get:
t ≈ 97.22 / 12 ≈ 8.1 years
Therefore, the closest answer to the options provided is 10 years (d).
Lastly, to verify our answer against a simple approximation, we can use the Rule of 70, which is an easy way to estimate the doubling time or halving time in our case. We can adjust this rule for the halving time by using 70 divided by the depreciation rate. In this case:
t ≈ 70 / 0.71 ≈ 98.59 months, which when divided by 12 is approximately 8.2 years, confirming our previous result.