Question
0 350
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A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)x
𝑓
(
𝑥
)
=
350
(
1.22
)
𝑥
can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)
Responses
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
1
2
3
4
5
A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)x
𝑓
(
𝑥
)
=
350
(
1.22
)
𝑥
can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)
Responses
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Answers
GPT-4o mini
To fill out the population chart of beetles using the equation \( f(x) = 350(1.22)^x \), we will calculate the number of beetles for each month \( x \) from 0 to 5.
1. For \( x = 0 \):
\[
f(0) = 350(1.22)^0 = 350(1) = 350
\]
2. For \( x = 1 \):
\[
f(1) = 350(1.22)^1 = 350(1.22) \approx 427
\]
3. For \( x = 2 \):
\[
f(2) = 350(1.22)^2 = 350(1.4884) \approx 521
\]
4. For \( x = 3 \):
\[
f(3) = 350(1.22)^3 = 350(1.8144) \approx 635
\]
5. For \( x = 4 \):
\[
f(4) = 350(1.22)^4 = 350(2.2198) \approx 777
\]
6. For \( x = 5 \):
\[
f(5) = 350(1.22)^5 = 350(2.7081) \approx 947
\]
Now, let's summarize the complete data in a chart format:
| Month (x) | Beetles (f(x)) |
|-----------|-----------------|
| 0 | 350 |
| 1 | 427 |
| 2 | 521 |
| 3 | 635 |
| 4 | 777 |
| 5 | 947 |
### Plot Points
The points to plot are:
- (0, 350)
- (1, 427)
- (2, 521)
- (3, 635)
- (4, 777)
- (5, 947)
### Graph Selection
The growth rate of 22% per month means that the graph of the function will be an increasing curve, not a decreasing one. Therefore, the correct graph to choose must show an increasing trend that best matches the calculated points.
Given the descriptions of the images you provided:
1. A decreasing curve with an arrow at the end.
2. An increasing curve with an arrow on the top.
3. A decreasing curve with an arrow at the end.
The correct response would be **the second graph (the increasing curve)**, as it reflects the nature of exponential growth in the population of beetles.
1. For \( x = 0 \):
\[
f(0) = 350(1.22)^0 = 350(1) = 350
\]
2. For \( x = 1 \):
\[
f(1) = 350(1.22)^1 = 350(1.22) \approx 427
\]
3. For \( x = 2 \):
\[
f(2) = 350(1.22)^2 = 350(1.4884) \approx 521
\]
4. For \( x = 3 \):
\[
f(3) = 350(1.22)^3 = 350(1.8144) \approx 635
\]
5. For \( x = 4 \):
\[
f(4) = 350(1.22)^4 = 350(2.2198) \approx 777
\]
6. For \( x = 5 \):
\[
f(5) = 350(1.22)^5 = 350(2.7081) \approx 947
\]
Now, let's summarize the complete data in a chart format:
| Month (x) | Beetles (f(x)) |
|-----------|-----------------|
| 0 | 350 |
| 1 | 427 |
| 2 | 521 |
| 3 | 635 |
| 4 | 777 |
| 5 | 947 |
### Plot Points
The points to plot are:
- (0, 350)
- (1, 427)
- (2, 521)
- (3, 635)
- (4, 777)
- (5, 947)
### Graph Selection
The growth rate of 22% per month means that the graph of the function will be an increasing curve, not a decreasing one. Therefore, the correct graph to choose must show an increasing trend that best matches the calculated points.
Given the descriptions of the images you provided:
1. A decreasing curve with an arrow at the end.
2. An increasing curve with an arrow on the top.
3. A decreasing curve with an arrow at the end.
The correct response would be **the second graph (the increasing curve)**, as it reflects the nature of exponential growth in the population of beetles.