To calculate the amount in the account after 6 years using the formula for continuous compounding, we will use the formula:
\[ A = P e^{rt} \]
Where:
- \(A\) is the balance after time \(t\),
- \(P\) is the principal amount (initial investment),
- \(e\) is the base of the natural logarithm (approximately 2.71828),
- \(r\) is the annual interest rate (as a decimal),
- \(t\) is the time in years.
From the problem:
- \(P = 700.00\)
- \(r = 0.06\) (since 6% = 0.06)
- \(t = 6\)
Now, substitute the values into the formula:
\[ A = 700 \cdot e^{0.06 \cdot 6} \]
Calculate the exponent first:
\[ 0.06 \cdot 6 = 0.36 \]
Now calculate \(e^{0.36}\):
Using a calculator, \(e^{0.36} \approx 1.4333\) (rounded to four decimal places).
Now substitute back:
\[ A = 700 \cdot 1.4333 \approx 1003.31 \]
Finally, rounding to the nearest cent:
\[ A \approx 1003.31 \]
Thus, the amount in the account after 6 years will be $1003.31.