Sure, let's solve the given problem step-by-step.
### Given Data:
- Principal (P): [tex]$10,000
- Rate (r): 2% (or 0.02 in decimal form)
- Compounded: annually (1 time per year)
- Time (t): 2 years
### Formula to Use:
The formula to calculate the amount of money accumulated after compounding interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = annual nominal interest rate (as a decimal).
- \( n \) = number of times the interest is compounded per year.
- \( t \) = the time the money is invested or borrowed for, in years.
### Step-by-Step Solution:
A. Finding the Amount After 2 Years:
1. Principal (P) = $[/tex]10,000
2. Rate (r) = 2% = 0.02 (as a decimal)
3. Compounded annually (n) = 1
4. Time (t) = 2 years
Plug these values into the formula:
[tex]\[ A = 10,000 \left(1 + \frac{0.02}{1}\right)^{1 \times 2} \][/tex]
[tex]\[ A = 10,000 \left(1 + 0.02\right)^2 \][/tex]
[tex]\[ A = 10,000 \times (1.02)^2 \][/tex]
[tex]\[ A = 10,000 \times 1.0404 \][/tex]
[tex]\[ A = 10,404.00 \][/tex]
So, the amount of money in the account after 2 years is [tex]$10,404.00
B. Finding the Interest Earned:
The interest earned is the total amount accumulated minus the principal.
\[ \text{Interest Earned} = A - P \]
\[ \text{Interest Earned} = 10,404.00 - 10,000 \]
\[ \text{Interest Earned} = 404.00 \]
So, the amount of interest earned is $[/tex]404.00.
### Final Answers:
A. The amount of money in the account after 2 years is [tex]$10,404.00
B. The amount of interest earned is $[/tex]404.00
