At the end of each month, for 24 months, $\$ 300$ is put into an account paying $8 \%$ annual interest compounded continuously. Find the future value of this account. Round your answer to the nearest cent.
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Step 1 ：Given that the deposits are made at the end of each month, this is a type of annuity, specifically an ordinary annuity, where payments are made at the end of each period.

Step 2 ：The future value of an ordinary annuity can be calculated using the formula: \(FV = P \cdot \left[\frac{e^{rt} - 1}{e^{r/n} - 1}\right] \cdot e^{r/n}\)

Step 3 ：Where: \n- \(FV\) is the future value of the annuity, \n- \(P\) is the amount of each payment, \n- \(r\) is the annual interest rate (in decimal), \n- \(t\) is the time the annuity is held for, in years, \n- \(n\) is the number of compounding periods per year.

Step 4 ：In this case, \(P = \$300\), \(r = 8\% = 0.08\), \(t = 24\) months = 2 years, and \(n = 12\) months/year.

Step 5 ：Substitute these values into the formula to calculate the future value: \n\(FV = 300 \cdot \left[\frac{e^{0.08 \cdot 2} - 1}{e^{0.08/12} - 1}\right] \cdot e^{0.08/12}\)

Step 6 ：Solving the above expression, we get \(FV \approx \$7834.04\)

Step 7 ：So, the future value of the account, after 24 months of depositing \$300 at the end of each month with an annual interest rate of 8% compounded continuously, is approximately \$7834.04.

Step 8 ：Final Answer: The future value of the account is approximately \(\boxed{\$7834.04}\)