Question

In 1626, Peter Minuit traded trinkets worth $24 for land on Manhattan Island. Assume that in 2014 the same land was worth $6 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.)

Answer 1

First compute how many years are there from 1626:
2014-1626= 388 years. 
Let the annual  rate be r. Then the formula is the following:
$24(1+r)^{388}=6,000,000,000$
We solve the above equation in order to find the value of r like this:
$(1+r)^{388}=\frac{6,000,000,000}{24}\\ 388 log(1+r)=\log(\frac{6,000,000,000}{24})\\ \log(1+r)=\log(\frac{6,000,000,000}{24})\times\frac{1}{388}\\ 1+r=\exp(\log(\frac{6,000,000,000}{24})\times\frac{1}{388})$
Using a calculator we get:
r=0.05
The annual rate is then 5%

Answer 2

The principal is
P = $24
Calculate the duration.
t = 2014 - 1626 = 388 years
The value after 388 years is
A = $6 x 10⁹
For continuous compounding, the compounding interval is
n = 365

Let r =  the rate.
Then use the formula
$P(1 + \frac{r}{n} )^{nt} = A$

That is,$24(1+ \frac{r}{365} )^{365*388} = 6 \times 10^{9}\\(1+ \frac{r}{365})^{ 141620} = \frac{6 \times 10^{9}}{24}= 2.5 \times 10^{8} \\1 + \frac{r}{365} =(2.5 \times 10^{8})^{1/141620} = 1.00013655$
Hence obtain
r/365 = 1.00013655 - 1 = 0.00013655
r = (0.00013655)*(365) = 0.0498 = 4.98%

Answer: 5.0%  (to 1 decimal place)