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Bot GPT 4-o mini To determine the correct graph that accurately displays Callie's investment situation, we need to analyze the function provided:
The function modeling her account balance is given as:
\[ a(t) = 1000(1.09)^t \]
Here, \( a(t) \) represents the amount in the account after \( t \) years, with the following conditions:
- The account starts with $1,000.
- It is compounded annually at a rate of 9%.
To find the value in the account after 10 years, we can calculate:
\[ a(10) = 1000(1.09)^{10} \]
Now, let's compute this value:
1. Calculate \( (1.09)^{10} \):
\[ (1.09)^{10} \approx 2.36736 \]
2. Multiply by $1,000:
\[ a(10) \approx 1000 \times 2.36736 \approx 2367.36 \]
So after 10 years, the amount in the account is approximately $2,367.36.
Now let’s compare this with the provided graph options:
1. The first option indicates ranges up to $3,500.
2. The second option only goes up to $2,000, which is not sufficient.
3. The third option goes up to $2,000 but provides an increment of 1 on the x-axis from 0 to 10.
4. The last option depicts a decreasing curve, which is incorrect since the function is increasing due to the compound interest.
Given the calculations and comparison of the values, the **third option,** which shows an increasing curve plotted from 0 to 10 years on the x-axis and an appropriate scale on the y-axis (up to $2,000) to likely include the value around $2,367.36, is the best choice.
Therefore, the correct response is:
**An increasing curve with an arrow at the end is plotted in the first quadrant of a coordinate plane. The x-axis ranges from 0 to 10 in increments of 1 and the y-axis ranges from 0 to 2,000 in increments of 500.**