Let's go through the step-by-step solution to solve for the amount that results when [tex]$4,000 is compounded at 6\% annually over seven years, and then calculate the interest earned in this case.
### Step-by-Step Solution:
1. Identify the known values:
- Principal (\(P\)): $[/tex]4,000
- Annual interest rate ([tex]\(r\)[/tex]): 6% (which needs to be converted to a decimal for calculations)
- Time ([tex]\(t\)[/tex]): 7 years
2. Convert the percentage rate to a decimal:
- [tex]\(0.06\)[/tex]
3. Use the compound interest formula:
[tex]\[
A = P(1 + r)^t
\][/tex]
Plugging in the values:
[tex]\[
A = 4000 \times (1 + 0.06)^7
\][/tex]
4. Perform the exponentiation and multiplication:
Calculate [tex]\((1 + 0.06)^7\)[/tex] first and then multiply by 4000.
Combining all the calculations:
[tex]\[
A = 4000 \times 1.5036305089913605
\][/tex]
[tex]\[
A = 6014.521035965442
\][/tex]
The amount that results when [tex]$4,000 is compounded at 6\% annually over seven years is \(\$[/tex]6,014.52\).
5. Calculate the interest earned:
[tex]\[
\text{Interest Earned} = \text{Final Amount} - \text{Principal}
\][/tex]
[tex]\[
\text{Interest Earned} = 6014.521035965442 - 4000
\][/tex]
[tex]\[
\text{Interest Earned} = 2014.521035965442
\][/tex]
The interest earned in this case is [tex]\(\$2,014.52\)[/tex].
### Summary:
- The amount that results when [tex]$4,000 is compounded at 6% annually over seven years: \(\$[/tex]6,014.52\)
- The interest earned in this case: [tex]\(\$2,014.52\)[/tex]
