At the end of an advertising​ campaign, weekly sales at an electronics store declined according to the equation y equals 9000 left parenthesis 3 Superscript negative 0.2 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
4750
9500
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 9,500 in increments of 950. From left to right, a curve falls steeply, passes through (10, 1,000), then falls shallowly, passing through (20, 111.1) and (30, 12.3). As x increases towards positive infinity, the curve approaches y=0 from above. All coordinates are approximate.
B.
0
30
60
0
4750
9500
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 9,500 in increments of 950. From left to right, a curve falls steeply, passes through (10, 3,000), then falls shallowly, passing through (20, 333.3) and (30, 37). As x increases towards positive infinity, the curve approaches y=0 from above. All coordinates are approximate.
C.
0
30
60
0
4750
9500
x
y

Answer 1

To answer your questions regarding the given function \( y = 9000 \cdot 3^{-0.2x} \):

Part 1: Graph the function

First, let's analyze the function to determine what a graph of it looks like. The function represents an exponentially decaying pattern based on the value of \( x \) (number of weeks since the advertising campaign ended).

1. The initial value \( y \) when \( x = 0 \) is: \[ y = 9000 \cdot 3^{0} = 9000 \cdot 1 = 9000 \text{ dollars} \]

2. As \( x \) increases, the factor \( 3^{-0.2x} \) will decrease, causing \( y \) to decline.

To see how the sales change over time, let's calculate a few values:

• When \( x = 10 \): \[ y = 9000 \cdot 3^{-0.2 \cdot 10} = 9000 \cdot 3^{-2} = 9000 \cdot \frac{1}{9} = 1000 \text{ dollars} \]
• When \( x = 20 \): \[ y = 9000 \cdot 3^{-4} = 9000 \cdot \frac{1}{81} \approx 111.1 \text{ dollars} \]
• When \( x = 30 \): \[ y = 9000 \cdot 3^{-6} = 9000 \cdot \frac{1}{729} \approx 12.3 \text{ dollars} \]

From the calculated points and knowing it's an exponential decay, you can now choose the appropriate graph.

Graph Selection: Based on the computations:

• The function starts at 9000 when \( x = 0 \), falls to 1000 when \( x = 10 \), and approaches 0 as \( x \) increases further.
  • Based on your descriptions, option A seems to align with these computed values best (the coordinates given correlate).

Part 2: Find weekly sales 10 weeks after the campaign ended

From our earlier calculation: At \( x = 10 \):

• Weekly sales = \( 1000 \) dollars.

Part 3: Comment on "It pays to advertise" for this store

This analysis shows that the sales figures significantly decrease after the advertising campaign ends, dropping from 9000 to 1000 dollars in just 10 weeks. This suggests that the impact of the advertising campaign had a pronounced effect on sales, and when the campaign ends, the sales also decline sharply.

Thus, it reinforces the idea that advertising is crucial for maintaining sales. Once the campaign concludes, the results sharply decline, illustrating the importance of ongoing marketing efforts to sustain sales levels. In essence, "It pays to advertise" for this store, as it directly correlates to maintaining higher sales figures.