In 1857 , a person sold a house to a lady for $\$ 26$. If the lady had put the $\$ 26$ into a bank account paying $4 \%$ interest, how much would the investment have been worth in the year 2009 if interest were compounded in the following ways?
a. monthly
b. continuously
a. If compounded monthly, the investment would be worth $\$ 11,248$ in 2009
(Round to the nearest dollar as needed.)
b. If compounded continuously, the investment would be worth $\$ \square$ in 2009
(Round to the nearest dollar as needed.)

Step 1 ：Given that the principal amount (P) is $26, the annual interest rate (r) is 4% or 0.04, and the time (t) is 2009 - 1857 = 152 years.

Step 2 ：For the case of monthly compounding, the number of times that interest is compounded per year (n) is 12.

Step 3 ：Using the formula for compound interest, A = P(1 + r/n)^(nt), we substitute the given values to get A = 26(1 + 0.04/12)^(12*152).

Step 4 ：Calculating the above expression, we get A = $11,248 (rounded to the nearest dollar).

Step 5 ：For the case of continuous compounding, we use the formula for continuous compound interest, A = Pe^(rt).

Step 6 ：Substituting the given values into the formula, we get A = 26e^(0.04*152).

Step 7 ：Calculating the above expression, we get A = $12,272 (rounded to the nearest dollar).

Step 8 ：Therefore, if the $26 were compounded continuously, it would be worth $12,272 in 2009.