At the end of an advertising​ campaign, weekly sales at an electronics store declined according to the equation y equals 3000 left parenthesis 3 Superscript negative 0.1 x Baseline right parenthesis ​dollars, where x is the number of weeks after the end of the campaign.

Question

a.
Graph the function for 0less than or equalsxless than or equals60.
b.
Use the graph to find the weekly sales 10 weeks after the campaign ended.
c.
Comment on​ "It pays to​ advertise" for this store.
Question content area bottom
Part 1
a. Choose the correct graph below.
A.
0
30
60
0
1750
3500
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 60 in increments of 5 and a vertical y-axis labeled from 0 to 3,500 in increments of 350. From left to right, a curve falls steeply, passes through (10, 500), then falls shallowly, passing through (20, 500) and (30, 500). As x increases towards positive infinity, the curve approaches y= 500 from above. All coordinates are approximate.
B.
0
30
60
0
1750
3500
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 60 in increments of 5 and a vertical y-axis labeled from 0 to 3,500 in increments of 350. From left to right, a curve falls steeply, passes through (10, 1,000), then falls shallowly, passing through (20, 333.3) and (30, 111.1). As x increases towards positive infinity, the curve approaches y=0 from above. All coordinates are approximate.
Your answer is correct.C.
0
30
60
0
1750
3500
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 60 in increments of 5 and a vertical y-axis labeled from 0 to 3,500 in increments of 350. From left to right, a curve falls steeply, passes through (10, 3,000), then falls shallowly, passing through (20, 1,000) and (30, 333.3). As x increases towards positive infinity, the curve approaches y=0 from above. All coordinates are approximate.
Part 2
b. The weekly sales 10 weeks after the campaign ended is ​$
  
enter your response here.

Answer 1

To solve the problem, we need to work through the steps you've outlined. The function you're given represents the weekly sales of an electronics store as a function of the number of weeks after the end of an advertising campaign:

\[ y = 3000 \times (3^{-0.1x}) \]

a. Graph the function for \( 0 \leq x \leq 60 \)

1. Function Overview: The function indicates that as time (in weeks) increases after the advertising campaign, the sales \( y \) will decline exponentially.

2. Behavior Analysis:

   • At \( x = 0 \): \[ y = 3000 \times (3^0) = 3000 \text{ dollars} \]
   • As \( x \to \infty \), \( y \) approaches \( 0 \), since \( 3^{-0.1x} \) decays toward zero.

3. Key Data Points:

   • Calculate \( y \) for a few key points. For example:
     • \( x = 10 \): \[ y = 3000 \times (3^{-1}) = 3000 \times \frac{1}{3} = 1000 \text{ dollars} \]
     • \( x = 20 \): \[ y = 3000 \times (3^{-2}) = 3000 \times \frac{1}{9} \approx 333.33 \text{ dollars} \]
     • \( x = 30 \): \[ y = 3000 \times (3^{-3}) = 3000 \times \frac{1}{27} \approx 111.11 \text{ dollars} \]

4. Graphing: The graph will start at 3000 dollars at \( x = 0 \), drop significantly as \( x \) rises, and asymptotically approach \( 0 \).

Given the graphical behavior and data points calculated, the correct choice represented from the options provided matches the description in part B:

• Graph C: shows the behavior correctly where it starts at 3000 and decreases sharply.

b. Calculate the sales 10 weeks after the campaign ended:

• We have already computed: \[ y(10) = 3000 \times 3^{-1} = 1000 \text{ dollars} \]

So, the weekly sales 10 weeks after the campaign ended is \($1000\).

c. Comment on "It pays to advertise" for this store.

Based on the provided model and calculations:

• The advertising campaign resulted in a high initial sales spike of $3000 immediately following the campaign. However, this model shows that the benefits of advertising are temporary, with sales declining to $1000 only 10 weeks later.
• Thus, while it initially pays to advertise by generating significant immediate revenue, the lasting impact appears minimal, as the sales trend suggests quickly diminishing returns over time after the campaign.
• In light of this analysis, it may be important for the store to consider strategies for sustaining sales after advertising, such as promotions, customer retention efforts, or ongoing marketing initiatives.