Answer:
The formula for the value of the machine after n years, denoted as v(n), can be found using the concept of exponential decay:
v(n) = Initial Value × (1 - Depreciation Rate)^n
In this case:
- Initial Value = $575,000
- Depreciation Rate = 10% or 0.10 (expressed as a decimal)
- n = number of years
Substitute these values into the formula:
v(n) = $575,000 × (1 - 0.10)^n
Simplified formula:
v(n) = $575,000 × 0.9^n
Explanation:
1. Start with the initial value of the machine: $575,000.
2. Apply the depreciation rate: 1 - 0.10 = 0.90 (since 10% is equivalent to 0.10).
3. Raise the value 0.90 to the power of n, representing the number of years.
4. Multiply the result by the initial value to get the value of the machine after n years.
So, the formula for the value of the machine after n years is v(n) = $575,000 × 0.9^n.