Chapter 2.3, Problem 10P

Suppose an amount $S_0$ is invested at an annual rate of return $r$ percent, compounded continuously.

Find the number of years $T$ that are required for the original amount to double in the value, as a function of $r$.

Determine $T$ if $r=8\%$.

Find the annual percentage rate that is needed for the original investment to double in $8$ years.

A rough guideline, known as early as the $15\text{th}$ century, is the “Rule of $72$”: an investment doubles when $rT$ is about $72$. How accurate is it for examples in $\left(b\right)$ and $\left(c\right)$? Explain the basis for the “Rule of $72$”. Why do you suppose the number $72$ was chosen long ago?