In 1626, Peter Minuit traded trinkets worth $24 for land on Manhattan Island. Assume that in 2014 the same land was worth $4 trillion. Find the annual rate of interest compounded continuously at which the $24 would have had to be invested during this time to yield the same amount. (Round your answer to one decimal place.)
4E12 = 24 * e^388r
ln(1.7E11) / 388 = r
To find the annual rate of interest compounded continuously, we can use the formula:
A = P * e^(rt)
Where:
A = final amount ($4 trillion)
P = initial principal ($24)
e = Euler's number (approximately 2.71828)
r = annual interest rate (what we need to find)
t = time period (1626 - 2014 = 388 years)
We want to solve for r, so by rearranging the equation:
r = ln(A/P) / t
Substituting the given values:
r = ln($4 trillion / $24) / 388
To calculate this using a scientific calculator or an online logarithm calculator, follow these steps:
1. Take the natural logarithm (ln) of the ratio $4 trillion / $24.
ln($4 trillion / $24) ≈ 24.90
2. Divide 24.90 by 388 to get the annual interest rate.
r ≈ 24.90 / 388 ≈ 0.0641
Rounding to one decimal place, the annual rate of interest compounded continuously is approximately 6.4%.