A $3\mathrm{.}15$ g marble is fired vertically upward using a spring gun. The spring must be compressed $7\mathrm{.}24$ cm if the marble is to just reach a target $29\mathrm{.}5$ m above the marble's position on the compressed spring. $($a$)$ What is the change AUg in the gravitational potential energy of the marble$-$Earth system during the $29\mathrm{.}5$ m ascent? $($b$)$ What is the change AUs in the elastic potential energy of the spring during its launch of the marble?$($c$)$ What is the spring constant of the spring?