A piece of machinery valued at $30,000 depreciates at a rate of 10% yearly. How long will it take for it to reach a value of $15,000?
$15k is half of $30k, so you want t years, where
0.9^t = 0.5
To find out how long it will take for the machinery to reach a value of $15,000, we can use the concept of depreciation.
Depreciation refers to the decrease in value of an asset over time due to factors like wear and tear, obsolescence, or market conditions. In this case, the machinery is depreciating at a rate of 10% per year.
To calculate the time it will take for the machinery to reach $15,000, we can set up an equation using the depreciation formula:
Value = Original Value * (1 - Depreciation Rate)^Time
Where:
- Value is the current value of the machinery,
- Original Value is the initial value of the machinery,
- Depreciation Rate is the rate at which the machinery depreciates per year,
- Time is the time period in years.
In this case, we want to find the time it will take for the machinery to reach a value of $15,000, so we can substitute the given values into the formula:
$15,000 = $30,000 * (1 - 0.10)^Time
Now, we can solve for Time. We can start by dividing both sides of the equation by $30,000:
$15,000 / $30,000 = (1 - 0.10)^Time
0.5 = 0.9^Time
Next, we can take the natural logarithm (ln) of both sides of the equation to solve for Time:
ln(0.5) = ln(0.9^Time)
Using logarithmic properties (ln(a^b) = b * ln(a)), we can simplify the equation:
ln(0.5) = Time * ln(0.9)
Now, divide both sides of the equation by ln(0.9):
ln(0.5) / ln(0.9) = Time
Using a calculator, we can evaluate the left side of the equation:
Time ≈ 7.37
Therefore, it will take approximately 7.37 years for the machinery to reach a value of $15,000, given a depreciation rate of 10% per year.