Certainly! Let's walk through the problem step by step to find how much money Miranda would have after investing [tex]$5,000 for three years with a compounded annual interest rate of 4%.
### Step-by-Step Solution:
1. Initial Investment:
Miranda starts with an initial investment (principal) of $[/tex]5,000.
2. Annual Interest Rate:
The annual interest rate is 4%, or 0.04 in decimal form.
3. Number of Years:
The investment period is 3 years.
4. Formula for Compound Interest:
To calculate the final amount with compound interest, we use the formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
where:
- [tex]\(A\)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\(P\)[/tex] is the principal amount (the initial money).
- [tex]\(r\)[/tex] is the annual interest rate (decimal).
- [tex]\(n\)[/tex] is the number of times that interest is compounded per year.
- [tex]\(t\)[/tex] is the number of years the money is invested.
In this case, since the interest is compounded annually, [tex]\(n = 1\)[/tex].
5. Plugging in the Values:
[tex]\[
A = 5000 \left(1 + 0.04\right)^3
\][/tex]
6. Calculate the Amount:
[tex]\[
A = 5000 \times (1.04)^3
\][/tex]
Let's break down the exponentiation:
[tex]\[
(1.04)^3 \approx 1.124864
\][/tex]
Now, multiply this result by the initial principal:
[tex]\[
A = 5000 \times 1.124864 \approx 5624.32
\][/tex]
7. Rounding to the Nearest Dollar:
The amount has been calculated as [tex]$5624.32. Since we need to round this to the nearest dollar, we get:
\[
\text{Rounded Amount} = 5624
\]
### Conclusion:
Miranda would have approximately $[/tex]5624 after investing [tex]$5000 for three years with a compounded interest rate of 4%.
Answer: $[/tex]5,624
