$4\mathrm{.}9$ Based on the dimensionless conduction heat rates for cases $12-15$ in Table $4\mathrm{.}16,$ find shape factors for the following objects having

temperature $T_1,$ located at the surface of a semi$-$infinite medium having temperature $T_2.$ The surface of the semi$-$infinite medium is adiabatic.

$($a$)$ A buried hemisphere, flush with the surface.

$($b$)$ A disk on the surface. Compare your result to Table $4\mathrm{.}19,$ case $10.$

$($c$)$ A square on the surface.

$($d$)$ A buried cube, flush with the surface.