Problem $6$: $(10$ points$)$ A $($perfectly cubical$)$ ice cube melts at a rate proportional to its surface area. That is$,$

$\backslash [$

$\backslash$frac$\{$d V$\}\{$d t$\}=-$k A

$\backslash ]$

where $\backslash ($ V $\backslash )$ is the volume of the ice cube, $\backslash ($ A $\backslash )$ is its surface area, and $\backslash ($ k $\backslash )$ is a positive constant.

a$.(2$ points$)$ Using the fact that $\backslash ($ V$=$s$^\{3\}\backslash )$ and $\backslash ($ A$=6$ s$^\{2\}\backslash )$ where $\backslash ($ s $\backslash )$ is the length of one of the cube's sides, translate the differential equation above into one involving only $\backslash ($ t $\backslash )($time$),\backslash ($ V $\backslash )($volume$),$ and $\backslash ($ k $\backslash )($the constant of proportionality$).$ That is$,$ eliminate the variable $\backslash ($ A $\backslash )$ by expressing $\backslash ($ A $\backslash )$ in terms of $\backslash ($ V $\backslash ).$

b$.(4$ points$)$ Solve the equation you found in $($a$)$ for a solution in terms of $\backslash ($ t $\backslash )($time$),\backslash ($ k $\backslash )($the constant$),$ and $\backslash ($ V$\_\{0\}\backslash )($the initial volume$).$ Hint: After integration, plug in $\backslash ($ t$=0\backslash )$ to rewrite $\backslash ($ C $\backslash )$ in terms of $\backslash ($ V$\_\{0\}\backslash ).$

c$.(2$ points$)$ Suppose the ice cube is initially $1$ centimeter cubed. If it takes $30$ seconds for the ice cube to melt, then what is $\backslash ($ k $\backslash )?$ Include units.

d$.(2$ points$)$ Continuing from part $($c$)$: After $15$ seconds will there be $($i$)$ more than half, $($ii$)$ less than half, or $($iii$)$ exactly half of the ice cube left?