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Select the correct answer.

The table shows the balance of an investment account at the beginning of each year the account was held. Assuming no other deposits have been made to the account, which statement describes the account's growth?

\begin{tabular}{|c|c|}
\hline Year & \begin{tabular}{c}
Account \\
Balance
\end{tabular} \\
\hline 1 & \[tex]$1,200.00 \\
\hline 2 & \$[/tex]1,260.00 \\
\hline 3 & \[tex]$1,323.00 \\
\hline 4 & \$[/tex]1,389.15 \\
\hline
\end{tabular}

A. The account is growing exponentially at an annual interest rate of [tex]$10.25\%$[/tex].
B. The account is growing linearly at an annual interest rate of [tex]$5.00\%$[/tex].
C. The account is growing exponentially at an annual interest rate of [tex]$500\%$[/tex].
D. The account is growing linearly at an annual interest rate of [tex]$10.25\%$[/tex].
E. The account is growing exponentially at an annual interest rate of [tex]$15.76\%$[/tex].



Answer :

GinnyAnswer
To determine how the account is growing, we need to analyze the given account balances at the beginning of each year. Let's break down the process step-by-step:

1. Identify the year-over-year growth rates:

- Year 1 to Year 2:
[tex]\[ \text{Growth Rate} = \frac{1260.00 - 1200.00}{1200.00} = \frac{60.00}{1200.00} = 0.05 = 5.00\% \][/tex]

- Year 2 to Year 3:
[tex]\[ \text{Growth Rate} = \frac{1323.00 - 1260.00}{1260.00} = \frac{63.00}{1260.00} \approx 0.05 = 5.00\% \][/tex]

- Year 3 to Year 4:
[tex]\[ \text{Growth Rate} = \frac{1389.15 - 1323.00}{1323.00} = \frac{66.15}{1323.00} \approx 0.05 = 5.00\% \][/tex]

2. Calculate the average growth rate:

Since all annual growth rates are the same:
[tex]\[ \text{Average Growth Rate} = \frac{5.00\% + 5.00\% + 5.00\%}{3} = 5.00\% \][/tex]

3. Determine the type of growth:

The consistent annual growth rate of 5.00% indicates that the account is growing linearly.

4. Choose the accurate statement:

Based on the computation:
[tex]\[ \text{Correct statement: The account is growing linearly at an annual interest rate of 5.00\%.} \][/tex]

So, the correct answer is:

[tex]\[ \boxed{B. \text{The account is growing linearly at an annual interest rate of } 5.00\%.} \][/tex]

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