To determine which graph accurately displays the situation described, we need to analyze the given equation for compound interest:
\[
a(t) = 1,000(1.09)^t
\]
Here, \( a(t) \) represents the amount in the account after \( t \) years, where the initial amount is $1,000 and the interest rate is 9%, compounded annually.
After 10 years, we can plug \( t = 10 \) into the equation to find out how much money Callie will have:
\[
a(10) = 1,000(1.09)^{10}
\]
Calculating \( (1.09)^{10} \):
\[
(1.09)^{10} \approx 2.36736
\]
Now calculating \( a(10) \):
\[
a(10) \approx 1,000 \times 2.36736 \approx 2367.36
\]
After 10 years, the amount accumulated in the account will be approximately $2,367.36.
Characteristics of the Graph
- Exponential Growth: The graph should show exponential growth, starting from $1,000 and increasing over time.
- End Value: At \( t = 10 \), the graph should reach around $2,367.36.
- Curve Shape: The curve should become steeper as \( t \) increases, indicating acceleration in growth due to compounding.
Choosing the Correct Graph
In an exam or a given set of graphs, you would look for a graph that reflects these characteristics—starting from $1,000, showing exponential growth, and crossing around $2,367.36 at the 10-year mark.
If you have specific graph options provided, you would select the one that meets these criteria. However, since the actual graphs are not displayed in this text, I can only guide you through the understanding of what to look for.
So, in summary, locate a graph showing exponential growth starting at $1,000 and reaching approximately $2,367.36 at \( t = 10 \).