You are considering investing in a fund that pays nominal annual rate of $7\mathrm{.}75\%,$ compounded quarterly. If you invest $$4,500$ at the end of each quarter, how long will it take for your account to grow to $$500,000?$

$14\mathrm{.}86$ years

$14\mathrm{.}22$ years

$14\mathrm{.}96$ years

$15\mathrm{.}10$ years

$15\mathrm{.}38$ years