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A machine depreciates by one fourth

of its value each year. If it costs​ $590 comma 000 ​new, what is its value after 6 ​years?
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Question content area right
Part 1
After 6 years the machine is worth ​$
  
enter your response here.
​(Simplify your answer. Round to the nearest dollar if​ needed.)

3 answers

To find the value of the machine after 6 years, we start with its initial cost of $590,000 and calculate its value at the end of each year, considering that it depreciates by one fourth (25%) each year.

This means that at the end of each year, the machine retains \(\frac{3}{4}\) (or 75%) of its value from the previous year.

The value of the machine after \(n\) years can be calculated using the formula:

\[ \text{Value after } n \text{ years} = \text{Initial Value} \times \left(\frac{3}{4}\right)^n \]

In this case:

  • Initial Value = $590,000
  • \(n = 6\)

Now we can calculate the value after 6 years:

\[ \text{Value after 6 years} = 590,000 \times \left(\frac{3}{4}\right)^6 \]

Calculating \(\left(\frac{3}{4}\right)^6\):

\[ \left(\frac{3}{4}\right)^6 = \frac{729}{4096} \approx 0.177978515625 \]

Now plug this into the value formula:

\[ \text{Value after 6 years} = 590,000 \times 0.177978515625 \approx 104,691.59 \]

Rounding to the nearest dollar, the value of the machine after 6 years is:

\[ \text{Value after 6 years} \approx 104,692 \]

So, the machine is worth $104,692 after 6 years.

wrong

Let’s carefully recalculate the value of the machine after 6 years, given that it depreciates by one fourth (25%) each year.

Initial Value

  • Initial cost of the machine: $590,000

Depreciation

As the machine depreciates by one fourth each year, it retains \( \frac{3}{4} \) of its value each year.

Formula

The value after \( n \) years can be calculated with the formula: \[ \text{Value after } n \text{ years} = \text{Initial Value} \times \left(\frac{3}{4}\right)^n \]

Calculation for 6 Years

  1. Calculate \( \left(\frac{3}{4}\right)^6 \): \[ \left(\frac{3}{4}\right)^6 = \frac{729}{4096} \approx 0.177978515625 \]

  2. Multiply by the initial value: \[ \text{Value after 6 years} = 590,000 \times 0.177978515625 \approx 104,691.59 \]

  3. Round to the nearest dollar: \[ \text{Value after 6 years} \approx 104,692 \]

This calculation confirms that the value of the machine after 6 years remains approximately $104,692.

If this answer is incorrect in a specific context or you're looking for a different approach or formula, please provide more details so I can assist you accurately.