A person places [tex] \$748 [/tex] in an investment account earning an annual rate of [tex] 8\% [/tex], compounded continuously. Using the formula [tex] V = P e^{rt} [/tex], where:

- [tex] V [/tex] is the value of the account in [tex] t [/tex] years,
- [tex] P [/tex] is the principal initially invested,
- [tex] e [/tex] is the base of the natural logarithm,
- [tex] r [/tex] is the rate of interest,

determine the amount of money, to the nearest cent, in the account after 4 years.



Answer :

To determine the amount of money in the investment account after 4 years, you can use the formula for continuous compounding:

[tex]\[ V = P e^{rt} \][/tex]

where:
- [tex]\( V \)[/tex] is the future value of the investment,
- [tex]\( P \)[/tex] is the initial principal (the amount of money initially invested),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a 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.

Let's break down the steps:

1. Identify the values of the variables in the problem:
- Initial principal, [tex]\( P \)[/tex] = \[tex]$748, - Annual interest rate, \( r \) = 8% = 0.08 (since percent needs to be converted to decimal form), - Time invested, \( t \) = 4 years. 2. Substitute these values into the formula: \[ V = 748 \cdot e^{0.08 \cdot 4} \] 3. Calculate the exponent: First, compute the product \( rt \): \[ 0.08 \cdot 4 = 0.32 \] Now the equation looks like: \[ V = 748 \cdot e^{0.32} \] 4. Compute \( e^{0.32} \): The value of \( e^{0.32} \) is a constant which we can approximate using a calculator: \[ e^{0.32} ≈ 1.377127764 \] 5. Multiply the initial principal by this value: \[ V = 748 \cdot 1.377127764 \] 6. Calculate the result: \[ V ≈ 1029.867172 \] 7. Round the result to the nearest cent: \[ V ≈ 1030.09 \] Therefore, after 4 years, the amount of money in the investment account would be approximately \$[/tex]1030.09.