A contract can be fulfilled by making an immediate payment of $\$ 11,000$ or equal payments at the end of each month for 12 years. What is the size of the monthly payments at $6.6 \%$ compounded monthly? The payment is $\$ \square$. (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Step 1 ：The problem is asking for the size of the monthly payments given a certain interest rate and a total amount. This is a typical problem of annuities in finance. The formula to calculate the monthly payment (PMT) for an annuity is: \(PMT = \frac{PV}{(1 - (1 + r)^{-n}) / r}\) where: PV is the present value, r is the monthly interest rate, and n is the total number of payments.

Step 2 ：Given that the present value (PV) is $11,000, the yearly interest rate is 6.6%, and the total number of payments is 12 years times 12 months.

Step 3 ：We first convert the yearly interest rate to a monthly interest rate by dividing it by 12. So, \(r_{monthly} = \frac{0.066}{12} = 0.0055\).

Step 4 ：Next, we calculate the total number of payments by multiplying the number of years by 12. So, \(n_{months} = 12 \times 12 = 144\).

Step 5 ：We can now plug these values into the formula to calculate the monthly payment: \(PMT = \frac{11000}{(1 - (1 + 0.0055)^{-144}) / 0.0055}\).

Step 6 ：Calculating the above expression, we get that the monthly payment is approximately $110.79.

Step 7 ：Final Answer: The size of the monthly payments is \(\boxed{\$110.79}\).