Chapter 1.1, Problem 97E

  $(a)$

To determine

To Find: Linear equation giving the value $\text{V}$ of the equipment for year $t$ during its $10$ years of use.

Answer to Problem 97E

  $\Rightarrow V=1011.5t+10995,0\le t\le 10$

Explanation of Solution

Given: Purchase price of equipment is $\$10,995$ and price of equipment after $10$ years is $\$880.$

Formula used: Equation of line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is:

  $y-y_1=(\frac{y_2-y_1}{x_2-x_1})(x-x_1)$

Calculation: We have given when $t=0$ , $V=10995$ and $t=10,V=880.$

So the corresponding points are $(t_1,v_1)=(0,10995)$ and $(t_2,v_2)=(10,880).$

Equation of line passing through $(t_1,v_1)$ and $(t_2,v_2)$ .

  $V-v_1=(\frac{v_2-v_1}{t_2-t_1})(t-t_1)$

  $\Rightarrow V-10995=(\frac{880-10995}{10-0})(t-0)$

  $\Rightarrow V-10995=(\frac{-10115}{10})t$

  $\Rightarrow V-10995=-1011.5t$

  $\Rightarrow V=-1011.5t+10995$

Also we have given $t$ is years between $0$ to $10$ .

Therefore, $0\le t\le 10.$

Conclusion:

Therefore, linear equation giving the value $\text{V}$ of the equipment for year $t$ during its $10$ years of use is: $V=-1011.5t+10995,0\le t\le 10.$

  $(b)$

To determine

To Graph: Graph the equation $V=-1011.5t+10995,0\le t\le 10$ using graphing utility.

To Verify: The answers algebraically.

Explanation of Solution

Given: $V=-1011.5t+10995,0\le t\le 10$

Graph:

  

By tracing points on graph we get the following values:

$t$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$V$	10995	$9983.5$	$8972$	$7960.5$	$6949$	$5937.5$	$4926$	$3914.5$	$2903$	$1891.5$	$880$

Algebraic Verification:

We have: $V=-1011.5t+10995,0\le t\le 10$

Substituting $0$ for $t$ we get:

  $V=-1011.5(0)+10995=10995$

Substituting $1$ for $t$ we get:

  $V=-1011.5(1)+10995=9983.5$

Substituting $2$ for $t$ we get:

  $V=-1011.5(2)+10995=8972$

Substituting $3$ for $t$ we get:

  $V=-1011.5(3)+10995=7960.5$

Substituting $4$ for $t$ we get:

  $V=-1011.5(4)+10995=6949$

Substituting $5$ for $t$ we get:

  $V=-1011.5(5)+10995=5937.5$

Substituting $6$ for $t$ we get:

  $V=-1011.5(6)+10995=4926$

Substituting $7$ for $t$ we get:

  $V=-1011.5(7)+10995=3914.5$

Substituting $8$ for $t$ we get:

  $V=-1011.5(8)+10995=2903$

Substituting $9$ for $t$ we get:

  $V=-1011.5(9)+10995=1891.5$

Substituting $10$ for $t$ we get:

  $V=-1011.5(10)+10995=880$

From these values table values are verified.

Conclusion:

Therefore, from graphical method and algebraic method we get the following table values:

$t$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$V$	10995	$9983.5$	$8972$	$7960.5$	$6949$	$5937.5$	$4926$	$3914.5$	$2903$	$1891.5$	$880$