Bot GPT 4-o mini To determine which graph accurately displays the amount in Callie's account after t years, we can analyze the function given, which is:
\[ a(t) = 1,000(1.09)^t \]
This represents an increasing exponential function since the amount in the account grows over time due to the 9% interest compounded annually.
Key Points:
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The account starts with $1,000 and grows at a rate of 9% annually.
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After 10 years, we need to calculate the amount in the account:
\[ a(10) = 1,000(1.09)^{10} \]
Performing the calculation:
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Calculate \( (1.09)^{10} \):
- \( (1.09)^{10} \approx 2.36736 \)
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Then calculate: \[ a(10) \approx 1,000 \times 2.36736 \approx 2,367.36 \]
This means that after 10 years, the amount in the account will be approximately $2,367.36.
Analyzing the Graphs:
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Graph 1:
- X-axis from 0 to 10 and Y-axis from 0 to 2,000.
- It shows an increasing curve that reaches above 2,000, indicating it could represent the correct growth.
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Graph 2:
- X-axis from 0 to 5 and Y-axis from 0 to 2,000.
- This graph does not extend long enough on the x-axis to cover 10 years.
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Graph 3:
- X-axis from 0 to 14 and Y-axis from 0 to 3,500.
- This graph also shows an increasing curve, which might extend beyond the 10-year mark and represents an increasing account balance.
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Graph 4:
- A decreasing curve, which does not apply to our situation since the account is increasing in value.
Conclusion:
Based on the calculations and analysis of the graphs, Graph 1 accurately displays the situation as it covers the 10-year time span and shows a growing amount that exceeds $2,000, consistent with the calculated growth of approximately $2,367.36.
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