Suppose that money is deposited daily into a savings account at an annual rate of $\$ 12,000$. If the account pays $5 \%$ interest compounded continuously, estimate the balance in the account at the end of 3 years.
The approximate balance in the account is \$ (Round to the nearest dollar as needed.)

Step 1 ：The problem is asking for the balance in the account after 3 years with continuous compounding. The formula for continuous compounding is \(A = P * e^{rt}\), where \(A\) is the amount of money accumulated after n years, including interest. \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal), and \(t\) is the time in years.

Step 2 ：In this case, the principal amount is not a lump sum but is deposited daily. So, we need to adjust the formula to account for this. We can do this by dividing the annual deposit by the number of days in a year to get the daily deposit and then summing up the compounded amount for each day's deposit over the course of the year.

Step 3 ：Given that the annual deposit is \$12000, the interest rate is 0.05, and the time is 3 years. There are 365 days in a year.

Step 4 ：Calculate the daily deposit: \(\frac{12000}{365} = 32.87671232876713\)

Step 5 ：Calculate the balance in the account at the end of 3 years: \(38838\)

Step 6 ：Final Answer: The approximate balance in the account at the end of 3 years is \(\boxed{38838}\)