Part A
we have the function
[tex]S(t)=\frac{1}{2}t+2[/tex]Remember that
The volume of a cube is given by the formula
[tex]V=S^3[/tex]substitute the function S(t) in the formula of volume
[tex]V(t)=(\frac{1}{2}t+2)^3[/tex]Part B
Find the surface area as the function of time
we have
[tex]SA(S)=6S^2[/tex]Find out (SAoS)(t)=SA(S(t))
so
[tex]SA(S(t))=6(\frac{1}{2}t+2)^2[/tex]Part C
we have
SA=216 in2
using the function of Part B
[tex]\begin{gathered} SA(t)=6(\frac{1}{2}t+2)^2 \\ \\ 6(\frac{1}{2}t+2)^2=216 \end{gathered}[/tex]Solve for t
[tex]\begin{gathered} (\frac{1}{2}t+2)^2=\frac{216}{6} \\ \\ (\frac{1}{2}t+2)^2=36 \\ take\text{ square root on both sides} \\ \\ \frac{1}{2}t+2=\pm6 \\ \\ \frac{1}{2}t=-2\pm6 \\ \\ t=-4\pm12 \end{gathered}[/tex]The values of t are
t=8 hours and t=-16 hours ( is not a solution because is a negative number)
therefore