Word Problem

Question

In 1626, Peter Minuit traded trinkets worth $24 for land on Manhattan Island. Assume that in 2014 the same land was worth $1 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.

Answer 1

1. Principal (P) =  the amount originally invested = $24
2. Rate (r)  = the interest rate  =  this is the unknown in the problem
3. Time (t) = the time period.  In this case 2014 - 1626 = 388 years
4. Amount (A) = the amount you would have at the end of the time period, i.e., in the future = $1Trillion
5. Equation for continuous compounding :  A= Pe^rt {Eqn 1}
6. Since we are looking for r in this problem, we need to rewrite Eqn 1, solving for r
1. A = Pe^rt
2. A/P = e^rt
3. take the log of both sides : ln(A/P) = ln (e^rt) = ln (A/P) = rt.  And finally ln (A/P) / t = r {Eqn 2}
7. Substitute the numbers into Eqn 2: 
1. 1,000,000,000,000 = 1 trillion = 1E12 (in scientific notation)
2. r =ln (1E12/24) / 388
3. r = 24.453 / 388
4. r = .0630
5. r = 6.30%
8. Pretty amazing, isn't it?  $24 turned into one trillion dollars. 

Answer 2

1 x 10¹² =  24e^{r•(2014-1626)} where I take trillion $ as ten to the 12^th bucks.

 

You can solve for r by taking the following steps & using a scientific calculator:

1. Divide both sides by 24

2. ln (left side value) = r(388)

3. one more step--divide by 388

 

If you wish to state the value as a %, shift the decimal place right 2 places and include %symbol.