To determine which statement accurately describes the account's growth, we will break down the process of calculating the yearly growth rate and then identify the nature of the growth based on these rates.
1. Start with the account balances for each year:
- Year 1: [tex]$1,200.00
- Year 2: $[/tex]1,260.00
- Year 3: [tex]$1,323.00
- Year 4: $[/tex]1,389.15
2. Calculate the yearly growth rate for each year:
- From Year 1 to Year 2: [tex]\[\text{Growth Rate} = \frac{1,260.00}{1,200.00} - 1 = 0.05\][/tex]
- From Year 2 to Year 3: [tex]\[\text{Growth Rate} = \frac{1,323.00}{1,260.00} - 1 = 0.05\][/tex]
- From Year 3 to Year 4: [tex]\[\text{Growth Rate} = \frac{1,389.15}{1,323.00} - 1 = 0.05\][/tex]
3. List the growth rates:
- Year 1 to Year 2: 0.05 (or 5%)
- Year 2 to Year 3: 0.05 (or 5%)
- Year 3 to Year 4: 0.05 (or 5%)
This gives us the growth rates: [tex]\([0.05, 0.05, 0.05]\)[/tex].
4. Calculate the average annual growth rate:
[tex]\[ \text{Average Annual Growth Rate} = \frac{0.05 + 0.05 + 0.05}{3} = 0.05 \][/tex]
The average annual growth rate is 0.05 (or 5%).
5. Determine the nature of the growth (linear or exponential):
- Each year, the growth rate is consistent at [tex]\(0.05\)[/tex] (5%), which implies a consistent rate compounded annually. This is characteristic of exponential growth.
6. Interpret the results and match with the correct option:
- Given the account is growing exponentially at a consistent annual growth rate of [tex]\(5.00\%\)[/tex], the correct statement is:
_C. The account is growing exponentially at an annual interest rate of [tex]$5.00 \%$[/tex]_
