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Suppose a business purchases equipment for ​$17 comma 000 and depreciates it over 5 years with the​ straight-line method until it reaches its salvage value of ​$2000 ​(see the figure​ below). Assuming that the depreciation can be for any part of a​ year, answer the questions to the

Write an equation that represents the depreciated value V as a function of the years t.
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​(Do not​ factor.)

1 answer

To calculate the depreciated value \( V \) of the equipment as a function of the years \( t \), we first need to determine the total depreciation over the 5-year period and the annual depreciation amount.

  1. Initial Cost of the Equipment: \( C = 17,000 \)
  2. Salvage Value: \( S = 2,000 \)
  3. Useful Life: \( n = 5 \) years

Step 1: Calculate the total depreciation

The total depreciation over the life of the equipment can be calculated as:

\[ \text{Total Depreciation} = C - S = 17,000 - 2,000 = 15,000 \]

Step 2: Calculate the annual depreciation

The annual depreciation is given by:

\[ \text{Annual Depreciation} = \frac{\text{Total Depreciation}}{n} = \frac{15,000}{5} = 3,000 \]

Step 3: Write the equation

The value \( V \) of the equipment after \( t \) years can thus be represented as:

\[ V(t) = C - (\text{Annual Depreciation} \times t) \]

Substituting in the values we calculated:

\[ V(t) = 17,000 - 3,000t \]

Final Result

So, the equation representing the depreciated value \( V \) as a function of the years \( t \) is:

\[ V(t) = 17,000 - 3,000t \]

This equation represents the value of the equipment at any point \( t \) during its useful life, from \( t = 0 \) to \( t = 5 \). After 5 years, the value \( V(5) \) will equal the salvage value of $2,000.