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.)
Just use the usual growth formula. It took 388 years, so
24e^(388r) = 4*10^12
To find the annual rate of interest compounded continuously, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Future value
P = Initial amount (principal)
e = Euler's number (approximately 2.71828)
r = Interest rate
t = Time in years
In this case, we need to find the interest rate (r) that would allow $24 to grow to $4 trillion over 388 years (2014 - 1626).
We can rearrange the formula to solve for r:
r = ln(A/P) / t
Where ln represents the natural logarithm.
Using the given values:
A = $4 trillion
P = $24
t = 388 years
Now, we can substitute these values into the formula and calculate the interest rate (r):
r = ln(4 trillion / 24) / 388
To evaluate this expression, we can use a calculator or a math software.
r ≈ 0.2393
Therefore, 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 is approximately 0.2393 or 23.9% (rounded to one decimal place).