To calculate the value of a machine that depreciates by one fourth of its value each year, we can use the formula for depreciation. The machine loses 25% of its value every year, which means it retains 75% of its value from the previous year.
Let's denote the initial value of the machine as \( V_0 = 510,000 \).
The value of the machine after \( t \) years, when it depreciates by 25% each year, can be calculated using the formula:
\[ V_t = V_0 \times (1 - d)^t \]
where \( d \) is the depreciation rate (in this case, \( d = 0.25 \)), and \( t \) is the number of years.
Now, substituting the values:
\[ V_t = 510,000 \times (1 - 0.25)^8 \] \[ = 510,000 \times (0.75)^8 \]
Now, we need to calculate \( (0.75)^8 \):
\[ (0.75)^8 \approx 0.100112915 \]
Now, multiply by the initial value:
\[ V_t = 510,000 \times 0.100112915 \approx 51,058.78 \]
Thus, the value of the machine after 8 years is approximately $51,058.78.
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