Use the expression [tex]P(1+r)^t[/tex] to solve the problems.

1. The amount that results when [tex]\$ 4,000[/tex] is compounded at [tex]6\%[/tex] annually over seven years: $\square[tex]$
2. The interest earned in this case: $[/tex]\square$



Answer :

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]