In 1859 , a person sold a house to a lady for $\$ 24$. If the lady had put the $\$ 24$ into a bank account paying $6 \%$ interest, how much would the investment have been worth in the year 2011 if interest were compounded in the following ways?
a. monthly
b. continuously

Step 1 ：First, we calculate the number of years from 1859 to 2011, which is 152 years.

Step 2 ：For part a, we use the formula for compound interest which is \(A = P(1 + \frac{r}{n})^{nt}\), where \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.

Step 3 ：Substituting the given values into the formula, we get \(A = 24(1 + \frac{0.06}{12})^{12*152}\), which simplifies to approximately \$214342.30.

Step 4 ：For part b, we use the formula for continuous compounding which is \(A = Pe^{rt}\), where \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate (in decimal), and \(t\) is the time the money is invested for in years.

Step 5 ：Substituting the given values into the formula, we get \(A = 24e^{0.06*152}\), which simplifies to approximately \$219268.84.

Step 6 ：Final Answer: \(\boxed{a. \$214342.30}\) If the interest were compounded monthly, the investment would have been worth approximately \$214342.30.

Step 7 ：Final Answer: \(\boxed{b. \$219268.84}\) If the interest were compounded continuously, the investment would have been worth approximately \$219268.84.