Question

Callie entered an art contest in second grade and won a $1,000 scholarship. The money was invested in an account paying a 9% interest rate compounded annually. The situation can be modeled by the equation a(t)=1,000(1.09)^t , where a is the amount in the account after t years. If Callie uses the scholarship 10 years later, determine which graph accurately displays the situation

Answer 1

To determine which graph accurately displays the situation, we first need to analyze the equation given:

\[ a(t) = 1,000(1.09)^t \]

In this equation:
- \( a(t) \) is the amount in the account after \( t \) years.
- The initial investment is $1,000.
- The interest rate is compounded annually at 9%, which means the amount grows exponentially.

Next, we will calculate \( a(10) \), which is the amount in the account after 10 years:

\[ a(10) = 1,000(1.09)^{10} \]

Now we need to calculate \( (1.09)^{10} \):

Using a calculator:

\[
(1.09)^{10} \approx 2.36736
\]

Thus,

\[
a(10) = 1,000 \times 2.36736 \approx 2367.36
\]

After 10 years, the amount in the account will be approximately $2,367.36.

When comparing the graphs, we should look for a graph that:
1. Starts at \( a(0) = 1,000 \) when \( t = 0 \).
2. Shows exponential growth as \( t \) increases, specifically steep growth around \( t = 10 \).
3. Reaches approximately \( a(10) \approx 2367.36 \) at \( t = 10 \).

The graph should be smooth and likely rises quickly since it's compounded annually at a fairly high interest rate.

If you provide a list of the graphs or a more detailed description, I can help you identify which one fits this description.