To determine which statement correctly describes the growth of the account based on the provided table, we need to calculate the growth rate of the account's balance from year to year. The table shows the account balances as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Year} & \text{Account Balance} \\
\hline
1 & \$1200.00 \\
\hline
2 & \$1260.00 \\
\hline
3 & \$1323.00 \\
\hline
4 & \$1389.15 \\
\hline
\end{array}
\][/tex]
### Step 1: Calculate the Yearly Growth Rates
To find the yearly growth rates, we use the formula for growth rate:
[tex]\[
\text{Growth Rate} = \frac{\text{Ending Balance} - \text{Beginning Balance}}{\text{Beginning Balance}}
\][/tex]
#### Year 1 to Year 2:
[tex]\[
\text{Growth Rate}_1 = \frac{1260.00 - 1200.00}{1200.00} = \frac{60.00}{1200.00} = 0.05 = 5\%
\][/tex]
#### Year 2 to Year 3:
[tex]\[
\text{Growth Rate}_2 = \frac{1323.00 - 1260.00}{1260.00} = \frac{63.00}{1260.00} \approx 0.05 = 5\%
\][/tex]
#### Year 3 to Year 4:
[tex]\[
\text{Growth Rate}_3 = \frac{1389.15 - 1323.00}{1323.00} = \frac{66.15}{1323.00} \approx 0.05 = 5\%
\][/tex]
### Step 2: Determine the Average Growth Rate
Since the growth rates are consistent each year, we can observe that the growth rate is approximately 5% each year.
### Step 3: Determine the Nature of Growth
From the given options, the account is either growing exponentially or linearly. Given that the growth rate is consistent at around 5% each year, this indicates exponential growth. In linear growth, the amount of increase would be constant, but with exponential growth, the percentage increase is constant.
### Conclusion
Based on the calculations and the options provided:
C. The account is growing exponentially at an annual interest rate of [tex]$5.00 \%$[/tex].
Thus, the correct answer is:
C. The account is growing exponentially at an annual interest rate of $5.00 \%.
