Q.20

Question

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, $b_{0}$ , and slope, $b_{1}$ , 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

Verified

Answer

(a) $y = 7200 - 1200 x .$

(b) $b_{0} = 7200$

$b_{1} = 1200$

(d) $y_{2} = 4800$

$y_{5} = 1200$

(e)

(f) $y_{4} = 2400$

1Part (a)Step 1: Given information

Given in the question that, 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.

We need to find the equation that expresses $y$ in terms of $x$

2Part(a) Step 2: Explanation

A computer system's initial worth is $$ 7200$ , and it decreases by $$ 1200$ per year for the next six years.

$y =$ The equipment's value.

$x =$ The equipment's age (in years).

After one year, the equipment is worth

$y = 7200 - 1200 \times 1 .$

After one year, the equipment is worth

$y = 7200 - 1200 \times 2 .$

As a result, the equipment's worth after $x (x = 1, 2, \dots 6)$ years is

$y = 7200 - 1200 x .$

3Part(b) Step 1: Given information

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.

We need to find the y-intercept, $b_{0}$ , and slope, $b_{1}$ , of the linear equation in part (a).

4Part (b) Step 2: Explanation

If the linear equation is of the form $y = b_{0} + b_{1} x$ , the $y$ -intercept is $b_{0}$ , and the slope is $b_{1}$ . The following is the calculated equation.

$y = 7200 + (- 1200) x$

As a result,

$b_{0} = 7200$

$b_{1} = 1200$

5Part(c) Step 1: Given information

We need to check that whether the line slopes upward, slopes downward, or is horizontal without graphing the equation in part (a).

6Part(c) Step 2: Explanation

For the linear equation $y = b_{0} + b_{1} x$ , the slope of the line is horizontal if $b_{1} = 0$ , upward if $b_{1} > 0$ , and downward if $b_{1} < 0$ .

The slope was discovered to be $- 1200$ . The graph of the linear equation $y = 7200 - 1200 x$ will therefore slope downward.

7Part(d) Step 1: Given information

We need to find the value of the computer equipment after $2$ years and also after $5$ years.

8Part (d) Step 2: Explanation

After two years, the worth of the computer is

$= 4800$

After five years, the worth of the computer is

$y_{5} = 7200 - 1200 \times 5$

$= 1200$

9Part(e) Step 1: Given information

We need to obtain the graph of the equation in part (a) by plotting the points from part (d) and connecting them with a line.

10Part (e) Step 2: Explanation

The graph of the equation $y = 7200 - 1200 x$ is shown below.

11Part(f) Step 1: Given information

We need to use the graph from part (e) to visually estimate the value of the equipment after $4$ years and also calculate that value exactly, using the equation from part (a).

12Part (f) Step 2: Explanation

The point $(4, 2400)$ is on the graph, and the value after four years is $$ 2400$ .

When applying the formula,

$y_{4} = 7200 - 1200 \times 4$

$= 2400$

Q. 19

Q. 20