Chapter 3.6, Problem 30E

a.

To determine

To find: an exponential model and a power model for the data and identify the coefficient of determination for each model using the regression feature of a graphing utility.

Answer to Problem 30E

Exponential model: $\mathrm{S}=117.1357{(2.4227)}^{\mathrm{t}},{\mathrm{r}}^2\approx \mathrm{0.95589.}$

Power model: $\mathrm{S}=148.345{\mathrm{t}}^{2.609},{\mathrm{r}}^2\approx 0.9983$

Explanation of Solution

Given information: The table shows the numbers of third-party skills S available for

Amazon’s Alexa in six consecutive three-month periods beginning with the first quarter

of 2016.

t	Skills, S
1	135
2	1,000
3	3,000
4	5,000
5	10,000
6	15,000

Calculation:

Using the graphing utility, the exponential model of the above table data is:

  $\mathrm{S}=117.1357{(2.4227)}^{\mathrm{t}},{\mathrm{r}}^2\approx \mathrm{0.95589.}$

Using the graphing utility, the power model of the above table data is:

  $\mathrm{S}=148.345{\mathrm{t}}^{2.609},{\mathrm{r}}^2\approx 0.9983$

b.

To determine

To graph: each model with the data using the graphing utility.

Answer to Problem 30E

Explanation of Solution

Given information: Exponential model: $\mathrm{S}=117.1357{(2.4227)}^{\mathrm{t}},{\mathrm{r}}^2\approx \mathrm{0.95589.}$

Power model: $\mathrm{S}=148.345{\mathrm{t}}^{2.609},{\mathrm{r}}^2\approx 0.9983$

Calculation:

The graph of Exponential model: $\mathrm{S}=117.1357{(2.4227)}^{\mathrm{t}},{\mathrm{r}}^2\approx \mathrm{0.95589.}$ is shown below.

  

The graph of Power model: $\mathrm{S}=148.345{\mathrm{t}}^{2.609},{\mathrm{r}}^2\approx 0.9983$ is shown below.

  

c.

To determine

To find: which model fits the data better.

Answer to Problem 30E

The power model fits the data better because ${\mathrm{r}}^2\approx 0.9983$ is close to 1 than ${\mathrm{r}}^2\approx 0.95589$ .

Explanation of Solution

Given information:

Calculation:

The power model fits the data better because ${\mathrm{r}}^2\approx 0.9983$ is close to 1 than ${\mathrm{r}}^2\approx 0.95589$ .