Problem $1.$

A $0\mathrm{.}5$ x $0\mathrm{.}5$ x $0\mathrm{.}5$ m$^3$box made of plywood k$=0\mathrm{.}12$ W$/$m$*$K is heated with a $100$W lamp, additionally a $5$W fan circulates the air causing internal convection h$=20$ W$/$m$^2*$K$.$ To regulate humidity a small humid air flow causes an average heat loss of $10$W$.$ Outside of the box the surrounding air has properties T$=20$ degrees C and h$=5$W$/$m$^2*$K$.$ Assume one dimensional heat transfer across each wall. Hint: The bottom of box is insulated, therefore assume identical dimensions and heat transfer across $4$ side walls and $1$ top wall. Ignore any radiation effects. Use thermal resistance method.

i$)$Find the correct wall thickness L to maintain an internal temperature of $37\mathrm{.}5$ degrees C$.$

ii$)$Recommend a second material from that can maintain $37\mathrm{.}5$ degrees C with a wall thickness maximum limit of $50$mm$.$