To determine the balance of the account after 5 years with an interest rate of 8.5% compounded continuously, we use the formula for continuous compounding. The formula 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 deposit).
- [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.
Given the values:
- [tex]\( P = \$ 1000 \)[/tex]
- [tex]\( r = 0.085 \)[/tex] (which is 8.5% expressed as a decimal)
- [tex]\( t = 5 \)[/tex] years
We substitute these values into the formula:
[tex]\[ A = 1000 \cdot e^{0.085 \cdot 5} \][/tex]
First, calculate the exponent:
[tex]\[ 0.085 \times 5 = 0.425 \][/tex]
So, the equation becomes:
[tex]\[ A = 1000 \cdot e^{0.425} \][/tex]
Using the natural constant [tex]\( e \)[/tex] (approximately 2.71828), we can calculate [tex]\( e^{0.425} \)[/tex]:
[tex]\[ e^{0.425} \approx 1.52959 \][/tex]
Now, multiply this value by the initial deposit [tex]\( P \)[/tex]:
[tex]\[ A = 1000 \cdot 1.52959 = 1529.59 \][/tex]
Thus, the balance after 5 years is approximately:
[tex]\[ \boxed{1529.59} \][/tex]
