To find the rate of depreciation, we can start with the depreciation equation: y = a(1 – r)^t, where:
- y is the current value of the car,
- a is the original cost of the car,
- r is the rate of depreciation, and
- t is time in years.
In this case, the current value of the car (y) is half of its original cost (a), so we can set up the equation as follows:
0.5a = a(1 – r)^t
Next, we divide both sides by a (assuming a is not zero):
0.5 = (1 – r)^t
At this point, we need to solve for r. Taking the natural logarithm of both sides gives:
ln(0.5) = ln((1 – r)^t)
Using the power rule of logarithms, this simplifies to:
ln(0.5) = t * ln(1 – r)
Now we can express ln(1 – r) in terms of ln(0.5):
ln(1 – r) = rac{ln(0.5)}{t}
To find the rate of depreciation r, we can rewrite this as:
1 – r = e^{ rac{ln(0.5)}{t}}
Thus:
r = 1 – e^{ rac{ln(0.5)}{t}} = 1 – 0.5^{ rac{1}{t}}
Now, depending on the time period t, you can substitute that value into the equation to find the specific rate of depreciation r. For example:
- If t = 1: r = 1 – 0.5 = 0.5 or 50%
- If t = 2: r = 1 – rac{1}{
oot{2}{2}} = 0.2929 or approximately 29.29%
In conclusion, the rate of depreciation will depend on the time period t. By substituting a specific value for t, you can determine the corresponding rate of depreciation.