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.