Chapter 2.3, Problem 11P

A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.

Determine the sum $S\left(t\right)$ accumulated at any time $t$.

If $r=5.5\%$, determine $k$ so that $\$1$ million will be available for retirement in $42$ years.

If $k=\$4,000/\text{year}$, determine the return rate $r$ that must be obtained to have $\$1$ million available in $42$ years.