Chapter 3.5, Problem 42E

a.

To determine

To find: a linear model for the value V of the tablet.

Answer to Problem 42E

Linear model is $V=-67.5t+300.$

Explanation of Solution

Given information: A new tablet computer that sold for $300in 2018 has a book value V

of $165 after 2 years .

Calculation:

Find the slope through the points (0, 300) and (2, 165).

  $m=\frac{300-165}{0-2}=-\mathrm{67.5.}$

The V −intercept is (0, 300). So the linear model is

  $V=-67.5t+300.$

b.

To determine

To find: an exponential model for the value V of the tablet and round the numbers in the model to four decimal places.

Answer to Problem 42E

  $\text{An}\text{exponential}\text{model}\text{is}V=300e^{-0.2989t}.$

Explanation of Solution

Given information: A new tablet computer that sold for $300in 2018 has a book value V

of $165 after 2 years .

Calculation:

Use the point (0, 300) to find a .

  $\begin{array}{c}V=a{e}^{kt}\\ 300=a{e}^{k(0)}\\ 300=a\\ So,\text{}V=300e^{kt}\\ \text{Use}\text{the}\text{point}\text{(2}\text{,}1\text{65)}\text{to}\text{find}k\\ V=300e^{kt}\\ 165=300e^{2k}\\ e^{2k}=\frac{165}{300}\\ \ln (e^{2k})=\ln \left(\frac{165}{300}\right)\\ k=\frac{1}{2}\ln \left(\frac{165}{300}\right)\\ k\approx -0.2989\\ \text{An}\text{exponential}\text{model}\text{is}V=300e^{-0.2989t}.\end{array}$

c.

To determine

To graph: the two models in the same viewing window.

Answer to Problem 42E

Explanation of Solution

Given information: Given models are $V=-67.5t+300$ , $V=300e^{-0.2989t}$

Calculation:

The graph of the two models in the same window is shown below.

  

d.

To determine

To find: which model represents a greater depreciation rate in the first year.

Answer to Problem 42E

The exponential model has a greater depreciation rate in the first year.

Explanation of Solution

Given information:

Calculation:

The exponential model has a greater depreciation rate in the first year.

e.

To determine

To find: for what years is the value of the tablet greater using the linear model, the exponential model.

Answer to Problem 42E

Using the graph, the linear model predicts a greater value for almost two full years

(0 < t < 2).

The exponential model predicts a greater value after about two years ( t $\ge$ 2).

Explanation of Solution

Given information:

Calculation:

Using the graph, the linear model predicts a greater value for almost two full years

(0 < t < 2).

The exponential model predicts a greater value after about two years ( t $\ge$ 2).