Certainly! Let's calculate the balance of [tex]$1000 deposited in an account with an 8.5% annual interest rate, compounded continuously, after 5 years. Here is the step-by-step solution:
1. Understand the given values:
- \( P \) (Principal amount) = $[/tex]1000
- [tex]\( r \)[/tex] (Annual interest rate as a decimal) = 0.085
- [tex]\( t \)[/tex] (Time in years) = 5
2. Identify the formula for continuous compounding:
The formula for continuous compounding is:
[tex]\[
A = P \cdot e^{rt}
\][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
3. Substitute the given values into the formula:
[tex]\[
A = 1000 \cdot e^{(0.085 \cdot 5)}
\][/tex]
4. Calculate the exponent:
[tex]\[
0.085 \cdot 5 = 0.425
\][/tex]
5. Calculate [tex]\( e^{0.425} \)[/tex]:
Using a calculator or software, we find:
[tex]\[
e^{0.425} \approx 1.52959
\][/tex]
6. Multiply the principal amount by the exponentiated value:
[tex]\[
A = 1000 \cdot 1.52959 = 1529.5904196633787
\][/tex]
7. Round the result to the nearest cent:
[tex]\[
A \approx 1529.59
\][/tex]
So, the balance in the account after 5 years, with continuous compounding at an 8.5% interest rate, is approximately $1529.59.
