Chapter 3.5, Problem 42E

a.

To determine

To calculate : the linear model for the value $V$ of the laptop

Answer to Problem 42E

The linear model for the value $V$ of the laptop is $-275t+1200$

Explanation of Solution

Given information : The question is

A new laptop computer that sold for $\$1200$ in $2014$ has a book value $\$650$ after $2$ years

Calculation :

Given two pints

  $\begin{array}{l}\left(t_1,V_1\right)\to \left(0,1200\right)\\ \left(t_2,V_2\right)\to \left(2,650\right)\end{array}$

Solve for the slope $m$

  $\begin{array}{l}m=\frac{v_2-v_1}{t_2-t_1}=\frac{650-1200}{2-0}=-275\\ V-V_1=m\left(t-t_1\right)\\ V-1200=-275\left(t-0\right)\\ V=-275t+1200\end{array}$

b.

To determine

To calculate : the exponential model for the value $V$ of the sedan

Answer to Problem 42E

The exponential model for the value $V$ of the laptop is $1200e^{-0.3066t}$

Explanation of Solution

Given information : The question is

A new laptop computer that sold for $\$1200$ in $2014$ has a book value $\$650$ after $2$ years

Calculation :

Using the exponential model

  $\begin{array}{l}y=a{e}^{-bt}\\ t=0,V=1200\\ 1200=a{e}^{-b\left(0\right)}\\ a=1200\\ t=2\\ y=650\\ 650=a{e}^{-b\left(2\right)}\\ \frac{650}{1200}=e^{-2b}\\ \ln \frac{650}{1200}=\ln e^{-2b}\\ \ln \frac{650}{1200}=-2b\ln e\\ b=\frac{\ln \frac{650}{1200}}{-2}\\ b\approx 0.3066\\ V=1200e^{-0.3066t}\end{array}$

c.

To determine

To use : the graphing utility to graph the two models

Explanation of Solution

Given information : The question is

A new laptop computer that sold for $\$1200$ in $2014$ has a book value $\$650$ after $2$ years

Graph : Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the data

Step 4: Press $\boxed{\text{GRAPH}}$ button to graph the function.

The graph is obtained as:

  

Interpretation:

From the above graph it is observed that the red line shows the function $V=-275t+1200$ and the blue line shows the function $V=1200e^{-0.3066t}$

d.

To determine

which model is represent a greater depreciation rate in first year

Answer to Problem 42E

The exponential model is represent a greater depreciation rate in first year

Explanation of Solution

Given information : The question is

A new laptop computer that sold for $\$1200$ in $2014$ has a book value $\$650$ after $2$ years

From the function it is observed that the exponential model is represent a greater depreciation rate in first year

e.

To determine

In what year the value of the sedan is greater using the linear and exponential model

Explanation of Solution

Given information : The question is

A new laptop computer that sold for $\$1200$ in $2014$ has a book value $\$650$ after $2$ years

From the graph the linear model is $0<t<2$ and the exponential model is $t\ge 2$