To solve for the future value [tex]\(A\)[/tex] using the continuous compound interest formula, we will proceed with the given values and follow each necessary step.
1. Identify the given values:
- Principal amount, [tex]\(P = \$8,000\)[/tex]
- Annual interest rate, [tex]\(r = 7.12\%\)[/tex]
- Time in years, [tex]\(t = 2\)[/tex] years
2. Convert the annual interest rate from percentage to decimal:
- [tex]\(r = 7.12\% = 0.0712\)[/tex]
3. Write down the continuous compound interest formula:
[tex]\[
A = P \cdot e^{rt}
\][/tex]
- Here, [tex]\(e\)[/tex] is the base of the natural logarithm (approximately 2.71828).
4. Substitute the given values into the formula:
[tex]\[
A = 8000 \cdot e^{(0.0712 \cdot 2)}
\][/tex]
5. Calculate the exponent [tex]\(rt\)[/tex]:
[tex]\[
rt = 0.0712 \times 2 = 0.1424
\][/tex]
6. Raise [tex]\(e\)[/tex] to the power of [tex]\(rt\)[/tex]:
[tex]\[
e^{0.1424} \approx 1.153038
\][/tex]
7. Multiply this result by the principal amount [tex]\(P\)[/tex]:
[tex]\[
A = 8000 \cdot 1.153038 = 9224.302171318775
\][/tex]
8. Round the result to two decimal places:
[tex]\[
A \approx 9224.30
\][/tex]
Therefore, the future value [tex]\(A\)[/tex] after 2 years, when [tex]\(\$8,000\)[/tex] is invested at an annual interest rate of [tex]\(7.12\%\)[/tex] compounded continuously, is approximately [tex]\(\$9,224.30\)[/tex].
