To solve the given problem, we need to calculate the amount of money in the account after the given number of years and the interest earned. Here’s the step-by-step process:
### Part (a): Find the Amount of Money After 3 Years
To find the amount of money in the account after 3 years, we will use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (initial amount of money),
- [tex]\( r \)[/tex] is the annual nominal interest rate (as a decimal),
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year,
- [tex]\( t \)[/tex] is the time the money is invested for in years,
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
Given:
- Principal [tex]\( P = \$7000 \)[/tex]
- Annual interest rate [tex]\( r = 0.006 \)[/tex] (0.6% as a decimal)
- Number of compounding periods per year [tex]\( n = 4 \)[/tex] (quarterly)
- Time [tex]\( t = 3 \)[/tex] years
Substituting these values into the formula:
[tex]\[ A = 7000 \left(1 + \frac{0.006}{4}\right)^{4 \times 3} \][/tex]
Calculate the term inside the parenthesis first:
[tex]\[ \frac{0.006}{4} = 0.0015 \][/tex]
Then:
[tex]\[ 1 + 0.0015 = 1.0015 \][/tex]
Next, raise this to the power of [tex]\( 4 \times 3 \)[/tex]:
[tex]\[ (1.0015)^{12} \][/tex]
After calculating this power and then multiplying by the principal:
[tex]\[ A \approx 7000 \times 1.01814 = 7127.04 \][/tex]
So, the amount of money in the account after 3 years is:
[tex]\[ \$7127.04 \][/tex]
### Part (b): Find the Interest Earned
To find the interest earned, subtract the initial principal from the amount of money in the account after 3 years:
[tex]\[ \text{Interest earned} = A - P \][/tex]
Substituting the known values:
[tex]\[ \text{Interest earned} = 7127.04 - 7000 \][/tex]
[tex]\[ \text{Interest earned} = 127.04 \][/tex]
Therefore, the amount of interest earned after 3 years is:
[tex]\[ \$127.04 \][/tex]
### Summary:
a. The amount of money in the account after 3 years is [tex]\(\$7127.04\)[/tex].
b. The amount of interest earned is [tex]\(\$127.04\)[/tex].
