Solved on Feb 24, 2024

Depreciate a $2,100 computer over 7 years using straight-line method. Find the book value function, domain, graph, value after 3 years, and when value is$ 600.

STEP 1

Assumptions
1. The initial cost of the computer is $2100.<br />2. The computer is depreciated using the straight-line method.<br />3. The useful life of the computer is 7 years.<br />4. The salvage value at the end of the 7 years is assumed to be$ 0 unless otherwise specified.
5. The book value $V$ is a linear function of the age $x$ of the computer.

STEP 2

The straight-line depreciation method calculates the depreciation expense by dividing the difference between the initial cost and the salvage value by the useful life of the asset.
$Depreciation\, expense = \frac{Initial\, cost - Salvage\, value}{Useful\, life}$

STEP 3

Since the salvage value is assumed to be $0, the depreciation expense per year is:
$Depreciation\, expense = \frac{2100 - 0}{7}$

STEP 4

Calculate the depreciation expense per year.
$Depreciation\, expense = \frac{2100}{7} = 300$

STEP 5

Now, we can write the linear function for the book value $V$ of the computer as a function of its age $x$ in years. The book value each year decreases by the depreciation expense.
$V(x) = Initial\, cost - (Depreciation\, expense \times x)$

STEP 6

Substitute the known values into the linear function.
$V(x) = 2100 - (300 \times x)$

STEP 7

Simplify the linear function.
$V(x) = 2100 - 300x$
This is the linear function that expresses the book value $V$ of the computer as a function of its age $x$ .

STEP 8

The domain of the function is the set of all possible values of $x$ that the function can take. Since $x$ represents the age of the computer in years and the computer is depreciated over 7 years, the domain of $x$ is from 0 to 7, inclusive.
$Domain\, of\, V(x): 0 \leq x \leq 7$

STEP 9

To graph the linear function, we need two points. We can use the points where $x=0$ and $x=7$ .
When $x=0$ (the computer is new):
$V(0) = 2100 - 300 \times 0 = 2100$
When $x=7$ (the end of the useful life):
$V(7) = 2100 - 300 \times 7 = 0$

STEP 10

Plot the points $(0, 2100)$ and $(7, 0)$ on a graph with the horizontal axis representing the age $x$ in years and the vertical axis representing the book value $V$ .

STEP 11

Draw a straight line through the points $(0, 2100)$ and $(7, 0)$ to represent the linear depreciation of the computer over time.

STEP 12

To find the book value of the computer after 3 years, we substitute $x=3$ into the linear function.
$V(3) = 2100 - 300 \times 3$

STEP 13

Calculate the book value after 3 years.
$V(3) = 2100 - 900 = 1200$
The book value of the computer after 3 years is $1200.

STEP 14

To find when the computer will have a book value of $600, we set$ V(x) = 600 $and solve for$ x$.
$600 = 2100 - 300x$

STEP 15

Subtract 2100 from both sides of the equation.
$600 - 2100 = -300x$

STEP 16

Simplify the equation.
$-1500 = -300x$

STEP 17

Divide both sides by $-300$ to solve for $x$ .
$x = \frac{-1500}{-300}$

STEP 18

Calculate the value of $x$ .
$x = 5$
The computer will have a book value of $600 after 5 years.

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Depreciate a $2,100 computer over 7 years using straight-line method. Find the book value function, domain, graph, value after 3 years, and when value is$ 600.

STEP 1

STEP 2

The straight-line depreciation method calculates the depreciation expense by dividing the difference between the initial cost and the salvage value by the useful life of the asset.
$Depreciation\, expense = \frac{Initial\, cost - Salvage\, value}{Useful\, life}$

STEP 3

Since the salvage value is assumed to be $0, the depreciation expense per year is:
$Depreciation\, expense = \frac{2100 - 0}{7}$

STEP 4

Calculate the depreciation expense per year.
$Depreciation\, expense = \frac{2100}{7} = 300$

STEP 5

Now, we can write the linear function for the book value $V$ of the computer as a function of its age $x$ in years. The book value each year decreases by the depreciation expense.
$V(x) = Initial\, cost - (Depreciation\, expense \times x)$

STEP 6

Substitute the known values into the linear function.
$V(x) = 2100 - (300 \times x)$

STEP 7

Simplify the linear function.
$V(x) = 2100 - 300x$
This is the linear function that expresses the book value $V$ of the computer as a function of its age $x$ .

STEP 8

The domain of the function is the set of all possible values of $x$ that the function can take. Since $x$ represents the age of the computer in years and the computer is depreciated over 7 years, the domain of $x$ is from 0 to 7, inclusive.
$Domain\, of\, V(x): 0 \leq x \leq 7$

STEP 9

To graph the linear function, we need two points. We can use the points where $x=0$ and $x=7$ .
When $x=0$ (the computer is new):
$V(0) = 2100 - 300 \times 0 = 2100$
When $x=7$ (the end of the useful life):
$V(7) = 2100 - 300 \times 7 = 0$

STEP 10

Plot the points $(0, 2100)$ and $(7, 0)$ on a graph with the horizontal axis representing the age $x$ in years and the vertical axis representing the book value $V$ .

STEP 11

Draw a straight line through the points $(0, 2100)$ and $(7, 0)$ to represent the linear depreciation of the computer over time.

STEP 12

To find the book value of the computer after 3 years, we substitute $x=3$ into the linear function.
$V(3) = 2100 - 300 \times 3$

STEP 13

Calculate the book value after 3 years.
$V(3) = 2100 - 900 = 1200$
The book value of the computer after 3 years is $1200.

STEP 14

To find when the computer will have a book value of $600, we set$ V(x) = 600 $and solve for$ x$.
$600 = 2100 - 300x$

STEP 15

Subtract 2100 from both sides of the equation.
$600 - 2100 = -300x$

STEP 16

Simplify the equation.
$-1500 = -300x$

STEP 17

Divide both sides by $-300$ to solve for $x$ .
$x = \frac{-1500}{-300}$

STEP 18

Calculate the value of $x$ .
$x = 5$
The computer will have a book value of $600 after 5 years.

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STEP 1

STEP 2

STEP 3

Since the salvage value is assumed to be $0, the depreciation expense per year is:Depreciation expense=2100−07Depreciation\, expense = \frac{2100 - 0}{7}Depreciationexpense=72100−0​

STEP 4

Calculate the depreciation expense per year.Depreciation expense=21007=300Depreciation\, expense = \frac{2100}{7} = 300Depreciationexpense=72100​=300

STEP 5

STEP 6

Substitute the known values into the linear function.V(x)=2100−(300×x)V(x) = 2100 - (300 \times x)V(x)=2100−(300×x)

STEP 7

Simplify the linear function.V(x)=2100−300xV(x) = 2100 - 300xV(x)=2100−300xThis is the linear function that expresses the book value VVV of the computer as a function of its age xxx.

STEP 8

STEP 9

STEP 10

Plot the points (0,2100)(0, 2100)(0,2100) and (7,0)(7, 0)(7,0) on a graph with the horizontal axis representing the age xxx in years and the vertical axis representing the book value VVV.

STEP 11

Draw a straight line through the points (0,2100)(0, 2100)(0,2100) and (7,0)(7, 0)(7,0) to represent the linear depreciation of the computer over time.

STEP 12

To find the book value of the computer after 3 years, we substitute x=3x=3x=3 into the linear function.V(3)=2100−300×3V(3) = 2100 - 300 \times 3V(3)=2100−300×3

STEP 13

Calculate the book value after 3 years.V(3)=2100−900=1200V(3) = 2100 - 900 = 1200V(3)=2100−900=1200The book value of the computer after 3 years is $1200.

STEP 14

To find when the computer will have a book value of 600,weset600, we set 600,wesetV(x) = 600andsolvefor and solve for andsolveforx$.600=2100−300x600 = 2100 - 300x600=2100−300x

STEP 15

Subtract 2100 from both sides of the equation.600−2100=−300x600 - 2100 = -300x600−2100=−300x

STEP 16

Simplify the equation.−1500=−300x-1500 = -300x−1500=−300x

STEP 17

Divide both sides by −300-300−300 to solve for xxx.x=−1500−300x = \frac{-1500}{-300}x=−300−1500​

STEP 18

Calculate the value of xxx.x=5x = 5x=5The computer will have a book value of $600 after 5 years.

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Since the salvage value is assumed to be $0, the depreciation expense per year is:
$Depreciation\, expense = \frac{2100 - 0}{7}$

Calculate the depreciation expense per year.
$Depreciation\, expense = \frac{2100}{7} = 300$

Substitute the known values into the linear function.
$V(x) = 2100 - (300 \times x)$

Simplify the linear function.
$V(x) = 2100 - 300x$
This is the linear function that expresses the book value $V$ of the computer as a function of its age $x$ .

Plot the points $(0, 2100)$ and $(7, 0)$ on a graph with the horizontal axis representing the age $x$ in years and the vertical axis representing the book value $V$ .

Draw a straight line through the points $(0, 2100)$ and $(7, 0)$ to represent the linear depreciation of the computer over time.

To find the book value of the computer after 3 years, we substitute $x=3$ into the linear function.
$V(3) = 2100 - 300 \times 3$

Calculate the book value after 3 years.
$V(3) = 2100 - 900 = 1200$
The book value of the computer after 3 years is $1200.

To find when the computer will have a book value of $600, we set$ V(x) = 600 $and solve for$ x$.
$600 = 2100 - 300x$

Subtract 2100 from both sides of the equation.
$600 - 2100 = -300x$

Simplify the equation.
$-1500 = -300x$

Divide both sides by $-300$ to solve for $x$ .
$x = \frac{-1500}{-300}$

Calculate the value of $x$ .
$x = 5$
The computer will have a book value of $600 after 5 years.