At the end of an advertising​ campaign, weekly sales at an electronics store declined according to the equation y equals 1000 left parenthesis 2 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
750
1500
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 1,500 in increments of 150. From left to right, a curve falls steeply, passes through (10, 1,000), then falls shallowly, passing through (20, 500) and (30, 250). As x increases towards positive infinity, the curve approaches y=0 from above. All coordinates are approximate.
B.
0
30
60
0
750
1500
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 1,500 in increments of 150. From left to right, a curve falls steeply, passes through (10, 500), then falls shallowly, passing through (20, 250) and (30, 125). As x increases towards positive infinity, the curve approaches y=0 from above. All coordinates are approximate.
C.
0
30
60
0
750
1500
x
y

Answer 1

To analyze the equation for weekly sales after an advertising campaign, we'll begin with the equation given:

\[ y = 1000 \cdot (2^{-0.1x}) \]

where \(y\) represents weekly sales in dollars and \(x\) represents the number of weeks after the advertising campaign ended.

Part (a): Graph the Function

1. Evaluation Points:

   • We can evaluate the function at several key integer points between 0 and 60 to get a good idea of what the graph looks like.
   • Let's calculate \(y\) for \(x = 0, 10, 20, 30, 40, 50, 60\):

   \[ x = 0: \quad y = 1000 \cdot (2^{0}) = 1000 \ x = 10: \quad y = 1000 \cdot (2^{-1}) = 500 \ x = 20: \quad y = 1000 \cdot (2^{-2}) = 250 \ x = 30: \quad y = 1000 \cdot (2^{-3}) = 125 \ x = 40: \quad y = 1000 \cdot (2^{-4}) = 62.5 \ x = 50: \quad y = 1000 \cdot (2^{-5}) = 31.25 \ x = 60: \quad y = 1000 \cdot (2^{-6}) = 15.625 \ \]

2. Graphing the Data Points:

   • The graph should show an exponential decay, starting at 1000 at \(x=0\) and approaching 0 as \(x\) increases towards 60.
   • The graph described is falling steeply initially and then flattens out as \(x\) increases.

3. Choose the Correct Graph:

   • The values we calculated show that at \(x=10\), \(y=500\), and so on. Hence, the graph that matches this description is Graph A.

Part (b): Find Weekly Sales 10 Weeks After

From our calculations in Part (a):

For \(x = 10\): \[ y = 500 \]

The weekly sales, 10 weeks after the campaign ended, are $500.

Part (c): Comment on "It Pays to Advertise"

• The equation shows that weekly sales decline exponentially after the advertising campaign ends. Even though the sales start at $1000, they drop significantly in just a few weeks.
• This indicates that the effectiveness of the advertising campaign has a limited lifespan. After 10 weeks, sales fall to $500, which is half of the initial sales.
• Therefore, "It pays to advertise" suggests that advertising has a substantial initial impact; however, once the advertising stops, the sales quickly decrease. This emphasizes the importance of continual marketing efforts to sustain sales rather than relying on a one-time campaign. Regular, ongoing marketing might be necessary to keep sales from declining significantly.

In summary, while advertising is beneficial, its effects diminish over time, underscoring the need for sustained or periodic marketing activities to maintain sales.