Question
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.
Vequals
enter your response here
(Do not factor.)
Write an equation that represents the depreciated value V as a function of the years t.
Vequals
enter your response here
(Do not factor.)
Answers
GPT-4o mini
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.
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.