Chapter 1, Problem 30RE

(a)

To determine

To find: The linear equation that gives the dollar value $\text{V}$ of the $\text{MP}3$ player in terms of the year $t$ .

Answer to Problem 30RE

The linear equation that gives the dollar value $\text{V}$ of the $\text{MP}3$ player in terms of the year $t$ is equal to $\text{V}\left(t\right)=285-47.50t$ .

Explanation of Solution

Given information:

The dollar value of an $\text{MP}3$ player in $2015$ is $\$285$ . The product will decrease in value at an expected rate of $\$47.50$ per year.

Calculation:

As the initial value of $\text{MP}3$ player in $2015$ i.e. $t=0$ is $\$285$ and the value of $\text{MP}3$ player decrease at an rate of $\$47.50$ per year.

The decrease in value of $\text{MP}3$ player for $t$ years is $\$47.50t$ and the value of $\text{MP}3$ player after $t$ years is equal to $285-47.50t$ .

Therefore, the linear equation that gives the dollar value $\text{V}$ of the $\text{MP}3$ player in terms of the year $t$ is equal to $\text{V}\left(t\right)=285-47.50t$ .

(b)

To determine

To graph: The equation found in part(a) in appropriate window for the graph and state its dimensions of the viewing window.

Answer to Problem 30RE

The dimensions of the viewing window are $\left[0,6\right]$ by $\left[0,285\right]$ as the intercepts for the equation are $6$ and $285$ .

Explanation of Solution

Given information:

The dollar value of an $\text{MP}3$ player in $2015$ is $\$285$ . The product will decrease in value at an expected rate of $\$47.50$ per year.

Calculation:

To graph a function $\text{V}\left(t\right)=285-47.50t$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=285-47.50t$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{285}\boxed{-}\boxed{47.50}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=285-47.50\text{X}$

Press the $\boxed{\text{WINDOW}}$ key and adjust the window by $\left[0,6\right]$ by $\left[0,285\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown in Figure (1).

  

Figure (1)

These dimensions are chosen as the intercepts formed by the line are $6$ and $285$ .

(c)

To determine

To find: The dollar value of $\text{MP}3$ player in $2019$ with the help of trace key and confirm the answer algebraically.

Answer to Problem 30RE

The dollar value of $\text{MP}3$ player in $2019$ is equal to $\$95$ .

Explanation of Solution

Given information:

The dollar value of an $\text{MP}3$ player in $2015$ is $\$285$ . The product will decrease in value at an expected rate of $\$47.50$ per year.

Calculation:

To graph a function $\text{V}\left(t\right)=285-47.50t$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=285-47.50t$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{285}\boxed{-}\boxed{47.50}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=285-47.50\text{X}$

Press the $\boxed{\text{WINDOW}}$ key and adjust the window by $\left[0,6\right]$ by $\left[0,285\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown in Figure (1).

Now enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{\text{1}}$ and then enter the value of $\text{X}=4$ as $\text{X}=4$ represents the year $2019$ .

  

Figure (1)

So, the dollar value of $\text{MP}3$ player in $2019$ is equal to $\$95$ .

Confirm the answer algebraically.

Substitute $4$ for $t$ in the function for value as $t=4$ represents the year $2019$ .

  $\begin{array}{l}\text{V}\left(t\right)=285-47.50t\\ \text{V}\left(4\right)=285-47.50\left(4\right)\\ \text{V}\left(4\right)=285-190\\ \text{V}\left(4\right)=95\end{array}$

Therefore, the dollar value of $\text{MP}3$ player in $2019$ is equal to $\$95$ .

(d)

To determine

To find: The year when the $\text{MP}3$ player will not have value.

Answer to Problem 30RE

The year when the $\text{MP}3$ player will not have value is $2021$ .

Explanation of Solution

Given information:

The dollar value of an $\text{MP}3$ player in $2015$ is $\$285$ . The product will decrease in value at an expected rate of $\$47.50$ per year.

Calculation:

As observed from the graph in part(b), the function of value touches the $x$ -axis at $x=6$ .

So, the $\text{MP}3$ player has $0$ value at $x=6$ .

The value $x=6$ corresponds to the year $2021$ .

Therefore, the year when the $\text{MP}3$ player will not have value is $2021$ .