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The principal represents an amount of money deposited in a savings account subject to compound interest at the given rate. Answer parts (a) and (b).

| Principal | Rate | Compounded | Time |
|-----------|------|------------|--------|
| [tex]$7000 | 0.6% | quarterly | 3 years|

a. Find how much money there will be in the account after 3 years.

The amount of money in the account after 3 years is $[/tex] _______.
(Round to the nearest cent as needed.)

b. Find the interest earned.

The amount of interest earned is $ _______.
(Round to the nearest cent as needed.)

Sagot :

GinnyAnswer
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].
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