Question

a famous painting was sold in 1945 for $22,170. in 1991 the painting was sold for $32.2 million. what rate of interest compounded continuously did this investment earn?
as an investment, the painting earned an interest rate of
(round to one decimal place as needed.)

Explanation:

Step1: Identify the formula for continuous - compounding

The formula for continuous - compounding is \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal amount, \(r\) is the interest rate, and \(t\) is the time.
Given \(P=\$22170\), \(A = \$3220000\), and \(t=1991 - 1945=46\) years.
Substitute the values into the formula: \(3220000=22170e^{46r}\).

Step2: Solve for \(r\)

First, divide both sides by \(22170\): \(\frac{3220000}{22170}=e^{46r}\).
\(\frac{3220000}{22170}\approx145.24\). So, \(145.24 = e^{46r}\).
Take the natural logarithm of both sides: \(\ln(145.24)=\ln(e^{46r})\).
Since \(\ln(e^{x})=x\), we have \(\ln(145.24) = 46r\).
\(\ln(145.24)\approx4.98\). Then \(r=\frac{4.98}{46}\).
\(r\approx0.108\) or \(r = 10.8\%\)

Answer:

\(10.8\%\)