Sure, let's go through this step-by-step using the formula for continuously compounded interest.
The formula to calculate the amount in an account with continuously compounded interest is:
[tex]\[ A = P \times e^{rt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (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),
- [tex]\( A \)[/tex] represents the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
Given:
- [tex]\( P = 900 \)[/tex] dollars,
- [tex]\( r = 0.07 \)[/tex] (which is 7\% expressed as a decimal),
- [tex]\( t = 6 \)[/tex] years.
First, we'll calculate the exponent part, [tex]\( rt \)[/tex]:
[tex]\[ rt = 0.07 \times 6 = 0.42 \][/tex]
Next, we raise [tex]\( e \)[/tex] to the power of [tex]\( rt \)[/tex]:
[tex]\[ e^{0.42} \approx 1.521 \][/tex]
Now, we can multiply this result by the initial principal amount:
[tex]\[ A = 900 \times 1.521 \approx 1369.77 \][/tex]
So, after rounding to the nearest cent, the amount in the account after 6 years is:
[tex]\[ \boxed{1369.77} \][/tex]
Thus, the balance after 6 years, with continuous compounding at an interest rate of 7%, will be approximately \$1369.77.
