Chapter 3.1, Problem 82E

a.

To determine

Draw the graph of the function.

Explanation of Solution

Given:

It is given in the question that a new sedan had a manufacturer’s suggested retail price of $\$31,340$ .After t years the sedan’s value is in the term of function $V(t)=31340{(\frac{4}{5})}^t$ .

Graph:

  

Interpretation: Here,the function $V(t)=31340{(\frac{4}{5})}^t$ shows the graph as above.It simply done by putting the equation on the graphing calculator and it will give the required graph.

b.

To determine

Find the value of V for $t=1$ to $t=10$ years by using graphing utility.

Answer to Problem 82E

The table is shown below:

t	$t=1$	$t=2$	$t=3$	$t=4$	$t=5$	$t=6$	$t=7$	$t=8$	$t=9$	$t=10$
V	$25072$	$20057.6$	$16046.08$	$12836.864$	$10269.4912$	$8215.592$	$6572.474$	$5257.979$	$4206.383$	$3365.106$

Explanation of Solution

Given: It is given in the question that a new sedan had a manufacturer’s suggested retail price of $\$31,340$ .After t years the sedan’s value is in the term of function $V(t)=31340{(\frac{4}{5})}^t$ .

Concept Used:

In this, use the concept of putting the value in the function $V(t)=31340{(\frac{4}{5})}^t$ ,calculate this in a very accurate way.

Calculation: Now , use the equation $V(t)=31340{(\frac{4}{5})}^t$

For $t=1$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(1)=31340{\left(\frac{4}{5}\right)}^1$

  $V(1)=25072$

For ,$t=2$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(2)=31340{\left(\frac{4}{5}\right)}^2$

  $V(2)=31340\left(\frac{16}{25}\right)$

  $V(2)=20057.6$

For ,$t=3$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(3)=31340{\left(\frac{4}{5}\right)}^3$

  $V(3)=31340\left(\frac{64}{125}\right)$

  $V(3)=16046.08$

For ,$t=4$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(4)=31340{\left(\frac{4}{5}\right)}^4$

  $V(4)=31340\left(\frac{256}{625}\right)$

  $V(4)=12836.864$

For ,$t=5$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(5)=31340{\left(\frac{4}{5}\right)}^5$

  $V(5)=31340\left(\frac{1024}{3125}\right)$

  $V(5)=10269.4912$

For ,$t=6$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(6)=31340{\left(\frac{4}{5}\right)}^6$

  $V(6)=31340\left(\frac{4096}{15625}\right)$

  $V(6)=8215.592$

For ,$t=7$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(7)=31340{\left(\frac{4}{5}\right)}^7$

  $V(7)=31340\left(\frac{16384}{78125}\right)$

  $V(7)=6572.474$

For ,$t=8$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(8)=31340{\left(\frac{4}{5}\right)}^8$

  $V(8)=31340\left(\frac{65536}{390625}\right)$

  $V(8)=5257.979$

For ,$t=9$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(9)=31340{\left(\frac{4}{5}\right)}^9$

  $V(9)=31340\left(\frac{262144}{1953125}\right)$

  $V(9)=4206.383$

For ,$t=10$

  $V(t)=31340{\left(\frac{4}{5}\right)}^t$

  $V(10)=31340{\left(\frac{4}{5}\right)}^{10}$

  $V(10)=31340\left(\frac{1048576}{9765625}\right)$

  $V(10)=3365.106$

Conclusion:

The table is :

t	$t=1$	$t=2$	$t=3$	$t=4$	$t=5$	$t=6$	$t=7$	$t=8$	$t=9$	$t=10$
V	$25072$	$20057.6$	$16046.08$	$12836.864$	$10269.4912$	$8215.592$	$6572.474$	$5257.979$	$4206.383$	$3365.106$

c.

To determine

Find the time at which the sedan have no value.

Answer to Problem 82E

It never be zero , but its value definitely lows.

Explanation of Solution

Given: It is given in the question that a new sedan had a manufacturer’s suggested retail price of $\$31,340$ .After t years the sedan’s value is in the term of function $V(t)=31340{(\frac{4}{5})}^t$ .

Concept Used:

In this,Use the concept of graph analyizing to find the value .

Calculation:

Here, the given graph is :

  

Here,it is easily seen that in the graph the curve will approach the x − axis but not cut it,so it concluded that the value of sedan will down (approaches very low) but never be zero because $y=0$ is a horizontal asymptotes in the above graph.

Conclusion:

It never be zero , but its value definitely lows.