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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 right.

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 right.
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 right.
Write an inequality using t as the variable that indicates that the depreciated value V of the equipment is less than ​$8000.

1 answer

To compute the depreciated value of the equipment \( V \) using the straight-line depreciation method, you can use the following formula:

\[ V = C - D \cdot t \]

Where:

  • \( C \) is the initial cost of the equipment ($17,000).
  • \( D \) is the annual depreciation expense.
  • \( t \) is the number of years that have passed.

First, we need to calculate the annual depreciation expense \( D \):

\[ D = \frac{C - \text{salvage value}}{\text{useful life}} = \frac{17000 - 2000}{5} = \frac{15000}{5} = 3000 \]

So, the annual depreciation expense \( D \) is $3,000.

Now, we can express the depreciated value \( V \) after \( t \) years as:

\[ V = 17000 - 3000 \cdot t \]

We want to find the inequality that indicates that \( V \) is less than $8,000:

\[ 17000 - 3000t < 8000 \]

Now, we can solve this inequality step-by-step:

  1. Subtract 17000 from both sides:

\[ -3000t < 8000 - 17000 \]

\[ -3000t < -9000 \]

  1. Divide by -3000 (remember to flip the inequality when dividing by a negative number):

\[ t > \frac{-9000}{-3000} \]

\[ t > 3 \]

Therefore, the inequality that indicates the depreciated value \( V \) of the equipment is less than $8,000 is:

\[ t > 3 \]

This means that the equipment will have a depreciated value of less than $8,000 after more than 3 years.