Chapter 3.6, Problem 33E

a.

To determine

To find: an exponential model for the data using the regression feature of the graphing utility.

Answer to Problem 33E

Exponential model: $\mathrm{S}=43.588{(1.0668)}^{\mathrm{t}}$ .

Explanation of Solution

Given information: The table shows the annual sales S (in billions of dollars) of Lowe’s

for the years from 2013 through 2017.

Year	Sales S (in billions of dollars)
2013	53.42
2014	56.22
2015	59.07
2016	65.02
2017	68.62

Calculation:

Let t represent the year, with t = 3 corresponding to 2013, then the above table can be written as:

Year	Sales S (in billions of dollars)
3	53.42
4	56.22
5	59.07
6	65.02
7	68.62

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

  $\mathrm{S}=43.588{(1.0668)}^{\mathrm{t}}$ .

b.

To determine

To rewrite: the model from part (a) as a natural exponential model.

Answer to Problem 33E

  $\mathrm{S}\approx 43.588{\mathrm{e}}^{0.0647\mathrm{t}}.$

Explanation of Solution

Given information: $\mathrm{S}=43.588{(1.0668)}^{\mathrm{t}}$ .

Calculation:

  $\begin{array}{l}$ S=43.588{(1.0668)}^t$\mathrm{S}=43.588{\mathrm{e}}^{{(\ln 1.0668)}^{\mathrm{t}}}\\ \mathrm{S}\approx 43.588{\mathrm{e}}^{0.0647\mathrm{t}}.\end{array}$

c.

To determine

To predict: the annual sales of Lowe’s in 2020 using natural exponential model and is the value reasonable, explain.

Answer to Problem 33E

According to the model, the annual sales in 2020 will be about $83.24 billion.

$83.24 billion appears to be reasonable prediction based on the trend of the data shown by the graph.

Explanation of Solution

Given information: $\mathrm{S}\approx 43.588{\mathrm{e}}^{0.0647\mathrm{t}}.$

Calculation:

For 2020, let t = 10 and find S.

  $\mathrm{S}\approx 43.588{\mathrm{e}}^{0.0647(10)}\approx \mathrm{83.24.}$

According to the model, the annual sales in 2020 will be about $83.24 billion.

$83.24 billion appears to be reasonable prediction based on the trend of the data shown by the graph.