Let's break down the problem step by step to find out what the balance on James's credit card will be one year from now and to determine whether this situation is an example of exponential growth or decay.
### Initial Setup:
1. Initial Balance: James has an initial credit card balance of [tex]$4,289.
2. Monthly Interest Rate: The interest rate charged by the credit card company is 5% per month.
### Calculations:
#### Monthly Compounded Interest:
Interest is compounded monthly, meaning that each month's interest is applied to the balance, becoming part of the principal for the next month.
#### Formula for Compound Interest:
The formula for compound interest when compounded monthly is:
\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n months, including interest.
- \( P \) is the principal amount (the initial balance).
- \( r \) is the annual interest rate (expressed as a decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the time the money is invested for in years.
Given that James's balance is compounded monthly:
- \( P = \$[/tex]4,289 \)
- [tex]\( r = 0.05 \times 12 = 0.60 \)[/tex] (since 5% per month corresponds to 60% annually)
- [tex]\( n = 12 \)[/tex] (compounded monthly)
- [tex]\( t = 1 \)[/tex] year
Since the interest rate is monthly, we can adjust our formula to reflect monthly compounding directly:
[tex]\[ A = 4289 \times (1 + 0.05)^{12} \][/tex]
Using this formula, we calculate the final balance after one year:
### Conclusion:
- The final balance after one year on James's credit card is approximately \$7702.43.
### Determining Exponential Growth or Decay:
- Exponential Growth: This occurs when a quantity increases over time, such as a growing balance due to compounded interest.
- Exponential Decay: This happens when a quantity decreases over time, which does not apply in this scenario.
Given the conditions of the problem, where the balance increases each month due to interest, this clearly represents an example of exponential growth.
Therefore, the situation James is experiencing can be classified as exponential growth.
