Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate of 10%. The second car depreciates at an annual rate of 15%. What is the approximate difference in the ages of the two cars?
1.7 ears
2.0 years
3.1 years
or 5.0 years
Two different cars each depreciate to 60% of their respective original values. The first car depreciates at an annual rate of 10%. The second car depreciates at an annual rate of 15%. What is the approximate difference in the ages of the two cars?
log0.900.6 - log0.850.6 = 1.7
or, since you probably don't have a button for log0.9x,
log(0.6)/log(0.9) - log(0.6)/log(0.85)
or even
log(0.6)(1/log(0.9)-1/log(0.85))
To find the approximate difference in the ages of the two cars, we need to calculate the number of years it takes for each car to depreciate to 60% of its original value.
Let's assume the original value of both cars is 100.
For the first car, which depreciates at an annual rate of 10%, we can set up the following equation:
100 * (1 - 0.10)^n = 60
Simplifying this equation, we get:
0.90^n = 0.60
Taking the logarithm of both sides to solve for n, we have:
n * log(0.90) = log(0.60)
n ≈ log(0.60) / log(0.90)
n ≈ 5.1
So it takes approximately 5.1 years for the first car to depreciate to 60% of its original value.
For the second car, which depreciates at an annual rate of 15%, we can set up a similar equation:
100 * (1 - 0.15)^n = 60
Simplifying and solving for n, we get:
n ≈ log(0.60) / log(0.85)
n ≈ 2.3
So it takes approximately 2.3 years for the second car to depreciate to 60% of its original value.
The approximate difference in the ages of the two cars is:
5.1 - 2.3 ≈ 2.8 years
Therefore, the correct answer is not provided among the given options.
To find the approximate difference in the ages of the two cars, we can first determine the number of years it takes for each car to depreciate to 60% of its original value.
Let's assume the original value of each car is 100 units.
For the first car with an annual depreciation rate of 10%, we can calculate the value after each year:
Year 1: 100 - (10% of 100) = 100 - 10 = 90
Year 2: 90 - (10% of 90) = 90 - 9 = 81
Year 3: 81 - (10% of 81) = 81 - 8.1 = 72.9
It takes approximately 3 years for the first car to depreciate to 60% of its original value.
Now let's calculate the number of years it takes for the second car with an annual depreciation rate of 15% to depreciate to 60% of its original value:
Year 1: 100 - (15% of 100) = 100 - 15 = 85
Year 2: 85 - (15% of 85) = 85 - 12.75 = 72.25
Year 3: 72.25 - (15% of 72.25) = 72.25 - 10.8375 = 61.4125
Year 4: 61.4125 - (15% of 61.4125) = 61.4125 - 9.212875 = 52.199625
It takes approximately 4 years for the second car to depreciate to 60% of its original value.
Therefore, the approximate difference in the ages of the two cars is 4 years - 3 years = 1 year.
So, the correct answer is 1.7 years.