A lakes is covered with ice 5 cm thick and the atmospheric temperature above the ice is $-10°C$. The rate at which the ice layer thickness increases (in cm/hour) is R/4. Then R is, Thermal conductivity of ice =0.005 cal $c{m}^{-1}{s}^{-1}{C}^{-1}$, density of ice 0.9g/cc and latent heat of fusion of ice 80 $\frac{cal}{g}$

Consider layer of ice that is xcm thick. just below the ice temperature is $0°C$  and above it the temperature is- $\theta °C$ .  

Let A be the area of the ice Layer. 

Heat conducted through ice in next dt time will be  $dQ=\frac{KA\theta }{x}dt$
If ice layer thickens by  $dx,$  the mass of ice frozen  in time  $dt$  will be $=pAdx$   
$\therefore dQ=\left(pAdx\right)L\Rightarrow \frac{kA\theta }{x}dt=pALdx$ 
$\Rightarrow \frac{dx}{dt}=\frac{k\theta }{xpL}=\frac{\left(0.005calc{m}^{-1}{s}^{-1}{C}^{-1}\right)\times (10°C)}{(5cm°)\left(0.9g\text{}c{m}^{-3}\right)\left(80ca{\lg }^{-1}\right)}=0.00014cm{s}^{-1}$ 
$=0.00014\times 3600cmh{r}^{-1}=0.5cmh{r}^{-1}$