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.
