Question

(a) What annual rate of interest, compounded continuously for 100 years, would have multiplied Benjamin Franklin's original capital by \(90\) ? (b) In Benjamin Franklin's estimate that the original 1000 pounds would grow to \(131,000\) in 100 years, he was using an annual rate of 5\(\%\) and compounding once each year. What rate of interest per year when compounded continuously for 100 years would multiply the original amount by 131 ?

Short answer

The annual interest rate for part (a), compounded continuously for 100 years, is given by \(\frac{1}{100} \ln(90)\). The rate of interest which, when compounded continuously for 100 years, would multiply the original amount by 131, is given by \(\frac{1}{100} \ln(131)\). The actual value of these expressions can be computed with a calculator to obtain numeric interest rates.

Step-by-step solution

Computing the interest rate for part (a)

Start by rearranging the formula to solve for \(r\) : \[r = \frac{1}{t} \ln(\frac{A}{P})\]. Given that \(A = 90P\), \(P = 1\), and \(t = 100\), plug these values into the formula to find \(r\). After these substitutions, the calculation becomes \[r = \frac{1}{100} \ln(\frac{90}{1}).\] The calculated rate \(r\) for part (a) can be found by computing this expression.

Computing the interest rate for part (b)

The calculation for part (b) is similar to part (a). Simply replace \(A\) with \(131\) : \[r = \frac{1}{100} \ln(\frac{131}{1})\]. The calculated rate \(r\) for part (b) can be found by computing this expression.