Let's analyze the account growth step by step.
1. Balance at the beginning of each year:
- Year 1: $200.00
- Year 2: $208.00
- Year 3: $216.32
2. Calculate the annual growth rates:
- Growth rate from Year 1 to Year 2: \((\frac{208.00}{200.00}) - 1 = 0.04\)
- Growth rate from Year 2 to Year 3: \((\frac{216.32}{208.00}) - 1 = 0.04\)
3. Average the growth rates:
[tex]\[
\text{Average growth rate} = \frac{0.04 + 0.04}{2} = 0.04
\][/tex]
4. Convert to percentage:
[tex]\[
\text{Average growth rate percentage} = 0.04 \times 100 = 4.00\%
\][/tex]
5. Check for exponential growth at 4.00% annual interest:
- Expected balance for Year 2: \(200.00 \times (1 + 0.04) = 208.00\)
- Expected balance for Year 3: \(208.00 \times (1 + 0.04) = 216.32\)
Since the actual balances match the expected balances:
- 208.00 matches 208.00 for Year 2
- 216.32 matches 216.32 for Year 3
This indicates the account is growing exponentially at an annual interest rate of 4.00%.
6. Conclusion:
- Given the calculations and balances matching perfectly for exponential growth at a 4.00% rate:
The correct statement is:
B. The account is growing exponentially at an annual interest rate of [tex]$4.00 \%$[/tex].
