Math / Algebra

QuestionSuppose that a company has just purchased a new computer for $\$ 2500$ . The company chooses to depreciate using the straight-line method for 5 years. (a) Write a linear function that expresses the book value V of the computer as a function of its age x . (b) What is the domain of the function found in part (a)? (c) Graph the linear function. (d) What is the book value of the computer after 2 years? (e) When will the computer have a book value of $\$ 500$ ? (a) The linear function is $V(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to figure out how the value of a computer goes down over time, make a graph of it, and then answer some questions about specific values. Watch out! Make sure to understand that "straight-line depreciation" means the value goes down by the same amount each year.
Also, remember that the computer's value can't go below zero!

STEP 2

1. Find the Depreciation
2. Define the Function
3. State the Domain
4. Graph the Function
5. Calculate 2-Year Value
6. Calculate Time to $\$500$

STEP 3

Alright, so the computer starts out costing $\$2500$ and it's going to be depreciated over 5 years.
This means we need to divide the initial value of $\$2500$ by 5 to find out how much it depreciates each year.

STEP 4

So, $\$2500$ divided by 5 is $\$500$ .
That's our annual depreciation!
This means the computer loses $\$500$ in value every year.

STEP 5

Now, we can write our linear function.
We start with the initial value of the computer, which is $\$2500$ , and then subtract the depreciation multiplied by the number of years, which we'll call $x$ .

STEP 6

So, our function is $V(x) = 2500 - 500 \cdot x$ .
This tells us the book value $V$ of the computer after $x$ years.

STEP 7

The domain of our function is the set of all possible values for $x$ .
Since we're talking about the number of years, $x$ can't be negative.
Also, the computer is fully depreciated after 5 years, so $x$ can't be greater than 5.

STEP 8

Therefore, the domain is $0 \le x \le 5$ .

STEP 9

To graph the function, we can plot two points.
When $x = 0$ (the initial time), $V(0) = 2500 - 500 \cdot 0 = 2500$ .
So, we have the point $(0, 2500)$ .

STEP 10

When $x = 5$ (after 5 years), $V(5) = 2500 - 500 \cdot 5 = 2500 - 2500 = 0$ .
So, we have the point $(5, 0)$ .

STEP 11

Now, we can draw a straight line connecting these two points.
This line represents the book value of the computer over time.

STEP 12

To find the book value after 2 years, we plug $x = 2$ into our function: $V(2) = 2500 - 500 \cdot 2 = 2500 - 1000 = 1500$ .

STEP 13

So, after 2 years, the computer's book value is $\$1500$ .

STEP 14

To find out when the computer will have a book value of $\$500$ , we set $V(x)$ equal to $500$ and solve for $x$ : $500 = 2500 - 500 \cdot x$ .

STEP 15

Subtract $2500$ from both sides: $-2000 = -500 \cdot x$ .

STEP 16

Divide both sides by $-500$ : $x = \frac{-2000}{-500} = 4$ .

STEP 17

So, the computer will have a book value of $\$500$ after 4 years.

STEP 18

(a) $V(x) = 2500 - 500 \cdot x$ (b) $0 \le x \le 5$ (c) A straight line connecting the points $(0, 2500)$ and $(5, 0)$ . (d) $\$1500$ (e) 4 years

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Studdy Solution

STEP 1

STEP 2

1. Find the Depreciation
2. Define the Function
3. State the Domain
4. Graph the Function
5. Calculate 2-Year Value
6. Calculate Time to $\$500$

STEP 3

Alright, so the computer starts out costing $\$2500$ and it's going to be depreciated over 5 years.
This means we need to divide the initial value of $\$2500$ by 5 to find out how much it depreciates each year.

STEP 4

So, $\$2500$ divided by 5 is $\$500$ .
That's our annual depreciation!
This means the computer loses $\$500$ in value every year.

STEP 5

Now, we can write our linear function.
We start with the initial value of the computer, which is $\$2500$ , and then subtract the depreciation multiplied by the number of years, which we'll call $x$ .

STEP 6

So, our function is $V(x) = 2500 - 500 \cdot x$ .
This tells us the book value $V$ of the computer after $x$ years.

STEP 7

The domain of our function is the set of all possible values for $x$ .
Since we're talking about the number of years, $x$ can't be negative.
Also, the computer is fully depreciated after 5 years, so $x$ can't be greater than 5.

STEP 8

Therefore, the domain is $0 \le x \le 5$ .

STEP 9

To graph the function, we can plot two points.
When $x = 0$ (the initial time), $V(0) = 2500 - 500 \cdot 0 = 2500$ .
So, we have the point $(0, 2500)$ .

STEP 10

When $x = 5$ (after 5 years), $V(5) = 2500 - 500 \cdot 5 = 2500 - 2500 = 0$ .
So, we have the point $(5, 0)$ .

STEP 11

Now, we can draw a straight line connecting these two points.
This line represents the book value of the computer over time.

STEP 12

To find the book value after 2 years, we plug $x = 2$ into our function: $V(2) = 2500 - 500 \cdot 2 = 2500 - 1000 = 1500$ .

STEP 13

So, after 2 years, the computer's book value is $\$1500$ .

STEP 14

To find out when the computer will have a book value of $\$500$ , we set $V(x)$ equal to $500$ and solve for $x$ : $500 = 2500 - 500 \cdot x$ .

STEP 15

Subtract $2500$ from both sides: $-2000 = -500 \cdot x$ .

STEP 16

Divide both sides by $-500$ : $x = \frac{-2000}{-500} = 4$ .

STEP 17

So, the computer will have a book value of $\$500$ after 4 years.

STEP 18

(a) $V(x) = 2500 - 500 \cdot x$ (b) $0 \le x \le 5$ (c) A straight line connecting the points $(0, 2500)$ and $(5, 0)$ . (d) $\$1500$ (e) 4 years

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Studdy Solution

STEP 1

STEP 2

1. Find the Depreciation2. Define the Function3. State the Domain4. Graph the Function5. Calculate 2-Year Value6. Calculate Time to $500\$500$500

STEP 3

Alright, so the computer starts out costing $2500\$2500$2500 and it's going to be depreciated over **5** years.This means we need to divide the **initial value** of $2500\$2500$2500 by **5** to find out how much it depreciates each year.

STEP 4

So, $2500\$2500$2500 divided by **5** is $500\$500$500.That's our **annual depreciation**!This means the computer loses $500\$500$500 in value every year.

STEP 5

Now, we can write our **linear function**.We start with the **initial value** of the computer, which is $2500\$2500$2500, and then subtract the **depreciation** multiplied by the number of years, which we'll call xxx.

STEP 6

So, our function is V(x)=2500−500⋅xV(x) = 2500 - 500 \cdot xV(x)=2500−500⋅x.This tells us the **book value** VVV of the computer after xxx years.

STEP 7

The **domain** of our function is the set of all possible values for xxx.Since we're talking about the number of years, xxx can't be negative.Also, the computer is fully depreciated after **5** years, so xxx can't be greater than **5**.

STEP 8

Therefore, the **domain** is 0≤x≤50 \le x \le 50≤x≤5.

STEP 9

To graph the function, we can plot two points.When x=0x = 0x=0 (the **initial time**), V(0)=2500−500⋅0=2500V(0) = 2500 - 500 \cdot 0 = 2500V(0)=2500−500⋅0=2500.So, we have the point (0,2500)(0, 2500)(0,2500).

STEP 10

When x=5x = 5x=5 (after **5 years**), V(5)=2500−500⋅5=2500−2500=0V(5) = 2500 - 500 \cdot 5 = 2500 - 2500 = 0V(5)=2500−500⋅5=2500−2500=0.So, we have the point (5,0)(5, 0)(5,0).

STEP 11

Now, we can draw a straight line connecting these two points.This line represents the **book value** of the computer over time.

STEP 12

To find the **book value** after **2** years, we plug x=2x = 2x=2 into our function: V(2)=2500−500⋅2=2500−1000=1500V(2) = 2500 - 500 \cdot 2 = 2500 - 1000 = 1500V(2)=2500−500⋅2=2500−1000=1500.

STEP 13

So, after **2** years, the computer's **book value** is $1500\$1500$1500.

STEP 14

To find out when the computer will have a **book value** of $500\$500$500, we set V(x)V(x)V(x) equal to 500500500 and solve for xxx: 500=2500−500⋅x500 = 2500 - 500 \cdot x500=2500−500⋅x.

STEP 15

Subtract 250025002500 from both sides: −2000=−500⋅x-2000 = -500 \cdot x−2000=−500⋅x.

STEP 16

Divide both sides by −500-500−500: x=−2000−500=4x = \frac{-2000}{-500} = 4x=−500−2000​=4.

STEP 17

So, the computer will have a **book value** of $500\$500$500 after **4** years.

STEP 18

(a) V(x)=2500−500⋅xV(x) = 2500 - 500 \cdot xV(x)=2500−500⋅x (b) 0≤x≤50 \le x \le 50≤x≤5 (c) A straight line connecting the points (0,2500)(0, 2500)(0,2500) and (5,0)(5, 0)(5,0). (d) $1500\$1500$1500 (e) **4** years

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1. Find the Depreciation
2. Define the Function
3. State the Domain
4. Graph the Function
5. Calculate 2-Year Value
6. Calculate Time to $\$500$

Alright, so the computer starts out costing $\$2500$ and it's going to be depreciated over 5 years.
This means we need to divide the initial value of $\$2500$ by 5 to find out how much it depreciates each year.

So, $\$2500$ divided by 5 is $\$500$ .
That's our annual depreciation!
This means the computer loses $\$500$ in value every year.

Now, we can write our linear function.
We start with the initial value of the computer, which is $\$2500$ , and then subtract the depreciation multiplied by the number of years, which we'll call $x$ .

So, our function is $V(x) = 2500 - 500 \cdot x$ .
This tells us the book value $V$ of the computer after $x$ years.

The domain of our function is the set of all possible values for $x$ .
Since we're talking about the number of years, $x$ can't be negative.
Also, the computer is fully depreciated after 5 years, so $x$ can't be greater than 5.

Therefore, the domain is $0 \le x \le 5$ .

To graph the function, we can plot two points.
When $x = 0$ (the initial time), $V(0) = 2500 - 500 \cdot 0 = 2500$ .
So, we have the point $(0, 2500)$ .

When $x = 5$ (after 5 years), $V(5) = 2500 - 500 \cdot 5 = 2500 - 2500 = 0$ .
So, we have the point $(5, 0)$ .

Now, we can draw a straight line connecting these two points.
This line represents the book value of the computer over time.

To find the book value after 2 years, we plug $x = 2$ into our function: $V(2) = 2500 - 500 \cdot 2 = 2500 - 1000 = 1500$ .

So, after 2 years, the computer's book value is $\$1500$ .

To find out when the computer will have a book value of $\$500$ , we set $V(x)$ equal to $500$ and solve for $x$ : $500 = 2500 - 500 \cdot x$ .

Subtract $2500$ from both sides: $-2000 = -500 \cdot x$ .

Divide both sides by $-500$ : $x = \frac{-2000}{-500} = 4$ .

So, the computer will have a book value of $\$500$ after 4 years.

(a) $V(x) = 2500 - 500 \cdot x$ (b) $0 \le x \le 5$ (c) A straight line connecting the points $(0, 2500)$ and $(5, 0)$ . (d) $\$1500$ (e) 4 years