a) By considering the heat balance between time t and t+dt of a slab of unit area at depth z and thickness dz, show that its temperature changes at a rate given by: at O'T ot Oz? b) Assume that the temperature at the soil-air interface varies with time as T(0,t)=To + asinwt and that heat flow in and out of the solid is only by conduction. Show that the temperature at depth z is: T(3:1) =T,+ a(-)sin(ex-3) D where D=2(x/6) a(z)= a(0)exp(-2/D) (you can start by differentiating left hand side of T(z,t) equation with respect to t and the right hand side twice with respect to z) c) For a typical soil, K = 0.3 x 10-6 m/s. Over a period of one year, calculate D and hence find the peak-peak variation of temperature at depths of 1m, 3m, and 5m for the cases To=15C, a(O) = 20C