Bot GPT 4-o mini To solve the population growth of the beetles, we apply the formula:
\[ f(x) = 350(1.22)^x \]
where:
- \( f(x) \) represents the population after \( x \) months,
- \( 350 \) is the initial population,
- \( 1.22 \) represents the 22% increase per month.
We can compute the population for the first few months (from \( x = 0 \) to \( x = 5 \)):
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For \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350 \]
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For \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350 \times 1.22 = 427 \]
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For \( x = 2 \): \[ f(2) = 350(1.22)^2 = 350 \times 1.4884 \approx 520 \]
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For \( x = 3 \): \[ f(3) = 350(1.22)^3 = 350 \times 1.8147 \approx 634 \]
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For \( x = 4 \): \[ f(4) = 350(1.22)^4 = 350 \times 2.2090 \approx 772 \]
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For \( x = 5 \): \[ f(5) = 350(1.22)^5 = 350 \times 2.6880 \approx 941 \]
Putting this data into a chart, we have:
| Months (x) | Population (f(x)) | |------------|---------------------| | 0 | 350 | | 1 | 427 | | 2 | 520 | | 3 | 634 | | 4 | 772 | | 5 | 941 |
Now, let's plot these points on a graph. From the table, we can see that the population is increasing, hence the graph should be an increasing curve.
Given the descriptions of the graphs:
- The first graph describes an increasing curve.
- The second and fourth graphs depict decreasing curves.
- The third graph also describes an increasing curve but is much larger in scale.
Considering the computed values (especially for smaller values of \( x \)), the first graph matches the increasing trend and lower scale of the beetle population, unlike the third graph which has a different range.
Therefore, the correct graph that displays the situation is the first one: an increasing curve with an arrow at the top.
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