How Much Would \$125 Invested At $8\%$ Interest Compounded Continuously Be Worth After 16 Years? Round Your Answer To The Nearest Cent.$A(t)=P \bullet E^{r T}$A. \$367.26 B. \$285.00 C. \$428.24 D. \$449.58
How Much Would $125 Invested at 8% Interest Compounded Continuously Be Worth After 16 Years?
Understanding Continuous Compounding
Continuous compounding is a type of interest calculation where the interest is compounded on an initial principal amount over a period of time, with the frequency of compounding occurring infinitely often in that time period. This type of compounding is often used in financial calculations, such as calculating the future value of an investment.
The Formula for Continuous Compounding
The formula for continuous compounding is given by:
A(t) = P * e^(rt)
Where:
- A(t) is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (in decimal form).
- t is the time the money is invested for, in years.
Calculating the Future Value of an Investment
To calculate the future value of an investment, we need to plug in the values of P, r, and t into the formula. In this case, we are given:
- P = $125 (the initial amount of money)
- r = 8% or 0.08 (the annual interest rate)
- t = 16 years (the time the money is invested for)
Plugging in the Values
Now, we can plug in the values into the formula:
A(16) = 125 * e^(0.08 * 16)
Simplifying the Expression
To simplify the expression, we can first calculate the value of 0.08 * 16:
0.08 * 16 = 1.28
Now, we can plug this value back into the expression:
A(16) = 125 * e^1.28
Evaluating the Expression
To evaluate the expression, we can use a calculator or a computer program to calculate the value of e^1.28:
e^1.28 ≈ 3.6052
Now, we can multiply this value by 125:
A(16) ≈ 125 * 3.6052 ≈ 449.565
Rounding the Answer
Finally, we need to round the answer to the nearest cent. Therefore, the final answer is:
$449.58
This is the amount of money that would be accumulated after 16 years, including interest, if $125 is invested at an annual interest rate of 8% compounded continuously.
Conclusion
In this article, we have calculated the future value of an investment using the formula for continuous compounding. We have plugged in the values of P, r, and t into the formula and simplified the expression to evaluate the value of e^1.28. Finally, we have rounded the answer to the nearest cent to get the final answer of $449.58.
Q&A: Continuous Compounding and Investment
Frequently Asked Questions
In this article, we will answer some frequently asked questions about continuous compounding and investment.
Q: What is continuous compounding?
A: Continuous compounding is a type of interest calculation where the interest is compounded on an initial principal amount over a period of time, with the frequency of compounding occurring infinitely often in that time period.
Q: What is the formula for continuous compounding?
A: The formula for continuous compounding is given by:
A(t) = P * e^(rt)
Where:
- A(t) is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (in decimal form).
- t is the time the money is invested for, in years.
Q: How do I calculate the future value of an investment using continuous compounding?
A: To calculate the future value of an investment using continuous compounding, you need to plug in the values of P, r, and t into the formula. You can then simplify the expression and evaluate the value of e^(rt) to get the final answer.
Q: What is the difference between continuous compounding and compound interest?
A: Continuous compounding and compound interest are both types of interest calculations, but they differ in the frequency of compounding. Compound interest is calculated at regular intervals, such as monthly or annually, while continuous compounding is calculated infinitely often in the time period.
Q: Is continuous compounding always the best option?
A: No, continuous compounding is not always the best option. It is typically used for long-term investments, such as retirement accounts or savings plans. For short-term investments, compound interest may be a better option.
Q: Can I use continuous compounding for investments with variable interest rates?
A: No, continuous compounding is typically used for investments with fixed interest rates. If the interest rate is variable, you may need to use a different type of interest calculation.
Q: How do I calculate the interest rate for continuous compounding?
A: To calculate the interest rate for continuous compounding, you need to divide the annual interest rate by 100 to convert it to a decimal. For example, if the annual interest rate is 8%, you would divide 8 by 100 to get 0.08.
Q: Can I use continuous compounding for investments with taxes?
A: Yes, you can use continuous compounding for investments with taxes. However, you will need to take into account the tax implications of the investment and adjust the interest rate accordingly.
Q: How do I calculate the future value of an investment with taxes using continuous compounding?
A: To calculate the future value of an investment with taxes using continuous compounding, you need to plug in the values of P, r, and t into the formula, and then adjust the interest rate to account for the taxes. You can then simplify the expression and evaluate the value of e^(rt) to get the final answer.
Conclusion
In this article, we have answered some frequently asked questions about continuous compounding and investment. We have covered topics such as the formula for continuous compounding, calculating the future value of an investment, and the difference between continuous compounding and compound interest. We hope this article has been helpful in answering your questions about continuous compounding and investment.