To find a linear function that represents the value [tex]\( V \)[/tex] of the copier as a function of [tex]\( t \)[/tex] years, we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] of the linear equation [tex]\( V = mt + b \)[/tex].
The given data shows the value of the copier over a period of 8 years:
- At [tex]\( t = 0 \)[/tex], [tex]\( V = 12000 \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( V = 10750 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( V = 9500 \)[/tex]
- At [tex]\( t = 3 \)[/tex], [tex]\( V = 8250 \)[/tex]
- At [tex]\( t = 4 \)[/tex], [tex]\( V = 7000 \)[/tex]
- At [tex]\( t = 5 \)[/tex], [tex]\( V = 5750 \)[/tex]
- At [tex]\( t = 6 \)[/tex], [tex]\( V = 4500 \)[/tex]
- At [tex]\( t = 7 \)[/tex], [tex]\( V = 3250 \)[/tex]
- At [tex]\( t = 8 \)[/tex], [tex]\( V = 2000 \)[/tex]
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
The slope [tex]\( m \)[/tex] of a linear function is calculated using the formula:
[tex]\[ m = \frac{V_{\text{final}} - V_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} \][/tex]
Using the values at [tex]\( t = 0 \)[/tex] and [tex]\( t = 8 \)[/tex]:
[tex]\[ V_{\text{initial}} = 12000 \][/tex]
[tex]\[ V_{\text{final}} = 2000 \][/tex]
[tex]\[ t_{\text{initial}} = 0 \][/tex]
[tex]\[ t_{\text{final}} = 8 \][/tex]
Substitute these values into the formula:
[tex]\[ m = \frac{2000 - 12000}{8 - 0} = \frac{-10000}{8} = -1250 \][/tex]
So the slope [tex]\( m \)[/tex] is [tex]\( -1250 \)[/tex].
### Step 2: Calculate the y-intercept [tex]\( b \)[/tex]
The y-intercept [tex]\( b \)[/tex] can be found using the point-slope form of a linear equation, which rearranges to:
[tex]\[ b = V_{\text{initial}} - m \cdot t_{\text{initial}} \][/tex]
Given that [tex]\( V_{\text{initial}} = 12000 \)[/tex] and [tex]\( t_{\text{initial}} = 0 \)[/tex]:
[tex]\[ b = 12000 - (-1250) \cdot 0 = 12000 \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] is [tex]\( 12000 \)[/tex].
### Step 3: Write the Linear Function
With the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] known, the linear function representing the value [tex]\( V \)[/tex] of the copier as a function of [tex]\( t \)[/tex] years is:
[tex]\[ V = -1250t + 12000 \][/tex]
This equation reflects how the value of the copier depreciates linearly over time.
