Use the periodic compound interest formula. How much money will you have in 6 years if you invest $\$ 11,000$ at a $3.8 \%$ annual rate of interest compounded quarterly? How much will you have if it is compounded monthly?

Step 1 ：Given that the principal amount (P) is $11,000, the annual interest rate (r) is 3.8% or 0.038 in decimal form, and the time (t) is 6 years.

Step 2 ：We will use the formula for compound interest, which is \(A = P (1 + \frac{r}{n})^{nt}\), where A is the amount of money accumulated after n years, including interest, n is the number of times that interest is compounded per year.

Step 3 ：For the first part of the question, we are asked to find the amount if the interest is compounded quarterly. This means n = 4. Substituting these values into the formula, we get \(A_{quarterly} = 11000 (1 + \frac{0.038}{4})^{4*6}\).

Step 4 ：Calculating the above expression, we find that \(A_{quarterly} = 13802.07693391619\).

Step 5 ：For the second part of the question, we are asked to find the amount if the interest is compounded monthly. This means n = 12. Substituting these values into the formula, we get \(A_{monthly} = 11000 (1 + \frac{0.038}{12})^{12*6}\).

Step 6 ：Calculating the above expression, we find that \(A_{monthly} = 13811.962066447975\).

Step 7 ：Rounding to two decimal places, we find that if the interest is compounded quarterly, you will have approximately \$13,802.08 in 6 years. If the interest is compounded monthly, you will have approximately \$13,811.96 in 6 years.

Step 8 ：\(\boxed{A_{quarterly} = \$13,802.08}\)

Step 9 ：\(\boxed{A_{monthly} = \$13,811.96}\)