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

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 + (r)/(n) )^(nt) = A`

That is,
`24(1+ (r)/(365) )^(365*388) = 6 * 10^(9)\\(1+ (r)/(365))^( 141620) = (6 * 10^(9))/(24)= 2.5 * 10^(8) \\1 + (r)/(365) =(2.5 * 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)

Answer 2

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:

`image`
Using a calculator we get:
r=0.05
The annual rate is then 5%