A young person with no initial capital invests $k$ dollars per year in a retirement account at an annual rate of return $0\mathrm{.}08.$ Assume that investments are made continuously and that the return is compounded continuously.

a$)$ Write a differential equation which models the rate of the change of the sum $S(t)$ with $t$ in years $($this will involve the parameter $k.$

$S^{\text{'}}=$

Note: Use $S$ rather than $S(t)$ since the latter confuses the computer.

b$)$ Use part a$)$ to determine a formula for the sum $S(t)--($this will involve the parameter $k)$:

$S(t)=$

c$)$ What value of $k$ will provide $3781000$ dollars in $46$ years?

$k=$