Question

suppose that in 1605, a man bought a diamond worth the equivalent of $13. suppose that the man had instead been able to put the equivalent of $13 in a bank at 3% interest compounded continuously. what would that $13 have been worth in 2005?
in 2005, the $13 would have been worth $
(do not round until the final answer. then round to the nearest dollar as needed.)

Explanation:

Step1: Calculate the time period

The time period \(t\) from 1605 to 2005 is \(2005 - 1605=400\) years.

Step2: Use the continuous - compounding formula

The formula for continuous compounding is \(A = Pe^{rt}\), where \(P=\$13\), \(r = 0.03\) (since \(3\%=0.03\)), and \(t = 400\).
Substitute the values into the formula: \(A=13e^{0.03\times400}\).
First, calculate \(0.03\times400 = 12\). Then \(A = 13e^{12}\).
We know that \(e^{12}\approx162754.791419\).
So \(A=13\times162754.791419\).

Answer:

\(A = 13\times e^{0.03\times400}=13\times e^{12}\approx13\times162754.791419\approx2115812\)