Question

A principal amount of $830 is placed in a savings account with an annual interest rate of 3.72% compounded continuously. What is the total account balance after 29 years?

Answer 1

The total account balance after 29 years, with continuous compounding, is approximately $2441.8.

The formula for the future value of an investment with continuous compounding is given by:

`A=P \cdot e^(r t)`

Where:

A is the future value of the investment/loan, including interest.

P is the principal amount (initial investment).

r is the annual interest rate (in decimal form).

t is the time the money is invested or borrowed for, in years.

e is the mathematical constant approximately equal to 2.71828.

In this case:

P = $830

r=0.0372 (3.72% converted to decimal)

t=29 years

Substitute these values into the formula:

`A=830 \cdot e^(0.0372 \cdot 29)`

Now, calculate the result:

`\begin{aligned}& A \approx 830 \cdot e^(1.0788) \\& A \approx 830 \cdot 2.942025 \\& A \approx 2441.8\end{aligned}`

Answer 2

The total account balance after 29 years with an annual interest rate of 3.72% compounded continuously is approximately $2441.69.

Step-by-step explanation:

To calculate the total account balance after 29 years with an annual interest rate of 3.72% compounded continuously, we use the formula for continuous compounding, which is A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (decimal), t is the time the money is invested for in years, and e is the base of the natural logarithm, approximately equal to 2.71828.

In this case: P = $830, r = 3.72/100 = 0.0372, t = 29.

Now, plug the values into the formula:

A = 830e0.0372*29

A = 830 * e1.0788

Calculate e1.0788 using a calculator:

A = 830 * 2.9418 (approximately)

A = $2441.69 (rounded to two decimal places)

Thus, the total account balance after 29 years will be approximately $2441.69.