To find out how much an investment will be worth after 10 years with continuous compounding, we use the formula for continuous compound interest:
[tex]\[ A = P \cdot e^{rt} \][/tex]
where:
- [tex]\(A\)[/tex] is the amount of money accumulated after n years, including interest,
- [tex]\(P\)[/tex] is the principal amount (the initial amount of money),
- [tex]\(r\)[/tex] is the annual interest rate (decimal),
- [tex]\(t\)[/tex] is the time the money is invested for, in years,
- [tex]\(e\)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given:
- Principal ([tex]\(P\)[/tex]) = [tex]$535
- Annual interest rate (\(r\)) = 6\% = 0.06
- Time (\(t\)) = 10 years
Step-by-step solution:
1. Identify the given values:
- \(P = 535\)
- \(r = 0.06\)
- \(t = 10\)
2. Substitute these values into the continuous compound interest formula:
\[ A = 535 \cdot e^{0.06 \cdot 10} \]
3. Calculate the exponent part:
\[ 0.06 \cdot 10 = 0.6 \]
4. Substitute this result back into the equation:
\[ A = 535 \cdot e^{0.6} \]
5. Find the value of \(e^{0.6}\). This value is approximately 1.82212.
6. Multiply the principal with the value of \(e^{0.6}\):
\[ A = 535 \cdot 1.82212 \]
7. Perform the multiplication:
\[ A \approx 974.83 \]
So, after 10 years, the investment will be worth approximately \$[/tex]974.83.
Among the given options, the correct answer is:
[tex]\[ \$ 974.83 \][/tex]
