Suppose You Deposit $600 Into An Account That Pays 5 % 5\% 5% Annual Interest, Compounded Continuously. How Much Will You Have In The Account In 4 Years? F ( T ) = A E R T F(t) = Ae^{rt} F ( T ) = A E R T A) $732.84 B) $635.62 C) $729.30 D) $4,433.43

Suppose you deposit $600 into an account that pays 5% annual interest, compounded continuously. How much will you have in the account in 4 years?

Understanding Continuous Compounding

Continuous compounding is a method of calculating interest on an investment where the interest is compounded on an instant-by-instant basis. This means that the interest is applied continuously, rather than at fixed intervals such as monthly or annually. The formula for continuous compounding is given by the equation:

$$f(t) = ae^{rt}$$

where:

• $a$ is the initial principal balance (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
• $e$ is the base of the natural logarithm, approximately equal to 2.71828

Calculating the Future Value of the Investment

To calculate the future value of the investment, we need to substitute the given values into the formula. The initial principal balance is $600, the annual interest rate is 5% or 0.05 in decimal form, and the time the money is invested for is 4 years.

$$f(4) = 600e^{0.05 \times 4}$$

Simplifying the Equation

To simplify the equation, we can first calculate the value of $0.05 \times 4$, which is equal to 0.2.

$$f(4) = 600e^{0.2}$$

Evaluating the Exponential Function

The exponential function $e^{0.2}$ can be evaluated using a calculator or by using the change of base formula. Using a calculator, we get:

$$e^{0.2} \approx 1.2214$$

Calculating the Future Value

Now that we have the value of $e^{0.2}$, we can substitute it back into the equation to calculate the future value of the investment.

$$f(4) = 600 \times 1.2214$$

Multiplying the Values

To calculate the future value, we multiply the initial principal balance by the value of $e^{0.2}$.

$$f(4) = 732.84$$

Conclusion

Therefore, after 4 years, the account will have approximately $732.84.

Comparison of Options

Comparing the calculated value with the given options, we can see that the correct answer is:

A) $732.84

The other options are incorrect, and the correct answer is the one that matches the calculated value.

Discussion

Continuous compounding is a powerful tool for calculating the future value of an investment. By using the formula for continuous compounding, we can calculate the future value of an investment with high accuracy. In this example, we used the formula to calculate the future value of an investment of $600 with an annual interest rate of 5% compounded continuously for 4 years. The calculated value of $732.84 is the correct answer, and it is the one that matches the given options.

Continuous Compounding Formula

The formula for continuous compounding is given by the equation:

$$f(t) = ae^{rt}$$

where:

• $a$ is the initial principal balance (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
• $e$ is the base of the natural logarithm, approximately equal to 2.71828

Continuous Compounding vs. Discrete Compounding

Continuous compounding is different from discrete compounding, where the interest is compounded at fixed intervals such as monthly or annually. Continuous compounding is a more accurate method of calculating interest, as it takes into account the interest earned on the interest itself. In contrast, discrete compounding assumes that the interest is compounded at fixed intervals, and it does not take into account the interest earned on the interest itself.

Advantages of Continuous Compounding

Continuous compounding has several advantages over discrete compounding. Some of the advantages of continuous compounding include:

• Higher accuracy: Continuous compounding is a more accurate method of calculating interest, as it takes into account the interest earned on the interest itself.
• Higher returns: Continuous compounding can result in higher returns on investment, as it takes into account the interest earned on the interest itself.
• Flexibility: Continuous compounding can be used to calculate the future value of an investment with high accuracy, regardless of the time period or interest rate.

Conclusion

In conclusion, continuous compounding is a powerful tool for calculating the future value of an investment. By using the formula for continuous compounding, we can calculate the future value of an investment with high accuracy. In this example, we used the formula to calculate the future value of an investment of $600 with an annual interest rate of 5% compounded continuously for 4 years. The calculated value of $732.84 is the correct answer, and it is the one that matches the given options.
Q&A: Continuous Compounding

Q: What is continuous compounding?

A: Continuous compounding is a method of calculating interest on an investment where the interest is compounded on an instant-by-instant basis. This means that the interest is applied continuously, rather than at fixed intervals such as monthly or annually.

Q: What is the formula for continuous compounding?

A: The formula for continuous compounding is given by the equation:

$$f(t) = ae^{rt}$$

where:

• $a$ is the initial principal balance (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
• $e$ is the base of the natural logarithm, approximately equal to 2.71828

Q: How does continuous compounding differ from discrete compounding?

A: Continuous compounding is different from discrete compounding, where the interest is compounded at fixed intervals such as monthly or annually. Continuous compounding is a more accurate method of calculating interest, as it takes into account the interest earned on the interest itself. In contrast, discrete compounding assumes that the interest is compounded at fixed intervals, and it does not take into account the interest earned on the interest itself.

Q: What are the advantages of continuous compounding?

A: Some of the advantages of continuous compounding include:

• Higher accuracy: Continuous compounding is a more accurate method of calculating interest, as it takes into account the interest earned on the interest itself.
• Higher returns: Continuous compounding can result in higher returns on investment, as it takes into account the interest earned on the interest itself.
• Flexibility: Continuous compounding can be used to calculate the future value of an investment with high accuracy, regardless of the time period or interest rate.

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 can use the formula:

$$f(t) = ae^{rt}$$

where:

• $a$ is the initial principal balance (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
• $e$ is the base of the natural logarithm, approximately equal to 2.71828

Q: What is the significance of the base of the natural logarithm (e) in continuous compounding?

A: The base of the natural logarithm (e) is a mathematical constant that is approximately equal to 2.71828. It is used in the formula for continuous compounding to calculate the future value of an investment.

Q: Can continuous compounding be used for investments with variable interest rates?

A: Yes, continuous compounding can be used for investments with variable interest rates. However, the interest rate must be expressed as a decimal value, and the formula for continuous compounding must be adjusted accordingly.

Q: What are some common applications of continuous compounding?

A: Some common applications of continuous compounding include:

• Savings accounts: Continuous compounding can be used to calculate the future value of a savings account with a fixed interest rate.
• Investments: Continuous compounding can be used to calculate the future value of an investment with a variable interest rate.
• Loans: Continuous compounding can be used to calculate the future value of a loan with a fixed interest rate.

Q: How can I use continuous compounding in real-world scenarios?

A: Continuous compounding can be used in a variety of real-world scenarios, including:

• Savings plans: Continuous compounding can be used to calculate the future value of a savings plan with a fixed interest rate.
• Investment portfolios: Continuous compounding can be used to calculate the future value of an investment portfolio with a variable interest rate.
• Loan calculations: Continuous compounding can be used to calculate the future value of a loan with a fixed interest rate.

Conclusion

Continuous compounding is a powerful tool for calculating the future value of an investment. By using the formula for continuous compounding, you can calculate the future value of an investment with high accuracy, regardless of the time period or interest rate. Whether you are saving for a specific goal or investing in a portfolio, continuous compounding can help you make informed decisions about your money.