Q. 20 - Elementary Statistics

Question

Equipment Depreciation. A small company has purchased a computer system for \(7200 and plans to depreciate the value of the equipment by \)1200 per year for 6 years. Let x denote the age of the equipment, in years, and y denote the value of the equipment, in hundreds of dollars.

a. Find the equation that expresses y in terms of x.

b. Find the y-intercept. h. and slope, by, of the linear equation in part (a).

c. Without graphing the equation in part (a), decide whether the line slopes upward, slopes downward, or is horizontal. d. Find the value of the computer equipment after 2 years: after

5 years. 

e. Obtain the graph of the equation in part (a) by plotting the points from part (d) and connecting them with a line.

f. Use the graph from part (e) to visually estimate the value of the equipment after 4 years. Then calculate that value exactly, using

the equation from part (a).

Step-by-Step Solution

Answer

Part ( a ) : 

The regression equation y in terms of x is:

y=72-12x

Part ( b ) :

The y-intercept, $b_0$= 72

The slope, $b_1$= - 12

Part ( c ) :

The slope, $b_1$ = -12(<0)

So, the slope is downward

Part ( d ) :

the value of the computer equipment after 2 years is $1200 

Part ( f ) :

 the estimated value of the equipment after 4 years is $2400 

Step 1. Given

A small company has purchased a microcomputer system for $7200 and plans to depreciate the value of the equipment by $1200 per year for 6 years.

That is,

The y-intercept is $7200

The slope is $1200

Let x denote the age of the equipment in years and y denote the value of the equipment in hundreds of dollars

Step 2. Part ( a )

The regression equation is $y = b_0 +b_{1}x$

Where.

$b_0$=by-intercept

$b_1$= slope.

x = independent variable.

y dependent variable.

The regression equation y in terms of x is:

y=72-12x

Step 3. Part ( b )

The regression equation is y = 72 - 12x

The y-intercept, $b_0$= 72

The slope, $b_1$= - 12

Step 4. Part ( c )

The linear equation $y = b_0 +b_{1}x$, slopes upward if $b_1$ > 0 and slopes downward if $b_1$ < 0 , and is horizontal if $b_1$= 0 , when $x_1$, is fixed at a specified value

The slope, $b_1$ = -12(<0)

So, the slope is downward

Step 5. Part ( d )

The value of the computer equipment after 2 years is

y=72-12x

  =72-12(2)

  =$\boxed{48}$

Therefore, the value of the computer equipment after 2 years is $4800

The value of the computer equipment after 5 years is

y=72-12x

 =72-12(5)

 = $\boxed{12}$

Therefore, the value of the computer equipment after 2 years is $1200

Step 6. Part ( e )

The graph of the equation y=72- underline 12. with the points (2, 48) and (5, 12) is given below

Step 7. Part ( f )

Using part (e) the graph of the value of the value of the equipment after 4 years is given below.

From the graph we can see that the estimated value of the equipment after 4 years is $2500

The estimated value of the equipment after 4 years is

y = 72-12x

   =72-12(4)

   = 24

Therefore the estimated value of the equipment after 4 years is $2400