To determine which graph accurately displays the situation modeled by the equation \( a(t) = 1,000(1.09)^t \), we need to analyze the growth of the investment over time and its expected value after 10 years.
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Understand the equation: The equation calculates the amount in the account after \( t \) years, where \( 1,000 \) is the initial investment, and \( 1.09 \) represents the compound interest factor (9% annual interest).
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Calculate \( a(10) \): \[ a(10) = 1,000(1.09)^{10} \]
Calculating \( (1.09)^{10} \):
- Using a calculator, \( (1.09)^{10} \approx 2.36736 \).
Now, calculate the total amount: \[ a(10) \approx 1,000 \times 2.36736 \approx 2367.36 \]
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Analysis of the Graphs:
- The graph in the first quadrant should show an increasing curve since the amount grows over time due to compound interest.
- We need to confirm that after 10 years, the amount is expected to be just above $2,300.
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Review the provided options:
- Graph 1: An increasing curve that ranges from 0 to 2,000 in increments of 500 (potentially too low for year 10).
- Graph 2: A decreasing curve (incorrect, as the amount is increasing).
- Graph 3: An increasing curve that ranges from 0 to 2000 in increments of 500 (also likely too low).
- Graph 4: An increasing curve that ranges from 0 to 3,500 in increments of 500 (can accommodate values above 2,300).
Given that we expect the amount to be around $2,367.36 after 10 years, the only graph that properly reflects an increasing investment over time and accommodates for this expected amount is Graph 4.