A leveling survey starts at a benchmark with a published elevation of $500\mathrm{.}00$ feet. The rod reading at a turning point $($TP$)$ located $300$ feet away is $4\mathrm{.}20$ feet. The back$-$sight reading at the benchmark is $3\mathrm{.}80$ feet. What is the elevation of the turning point $($TP$)?$

${500\mathrm{.}40}^{\text{'}}$

${499\mathrm{.}60}^{\text{'}}$

$503\mathrm{.}80$

$504\mathrm{.}20$