## Question

###### Ground Temperature Mathematical models of ground temperature variation usually involve Fourier series or other sophisticated methods. However, the elementary model $$ u(x, t)=T_{0}+A_{0} e^{-a x} \cos \left(\frac{\pi}{6} t-a x\right) $$ has been developed for temperature $u(x, t)$ at a given location at a variable time $t$ (in months) and a variable depth $x$ (in centimeters) beneath Earth's surface, $T_{0}$ is the annual average surface temperature, and $A_{0}$ is the amplitude of th

Ground Temperature Mathematical models of ground temperature variation usually involve Fourier series or other sophisticated methods. However, the elementary model $$ u(x, t)=T_{0}+A_{0} e^{-a x} \cos \left(\frac{\pi}{6} t-a x\right) $$ has been developed for temperature $u(x, t)$ at a given location at a variable time $t$ (in months) and a variable depth $x$ (in centimeters) beneath Earth's surface, $T_{0}$ is the annual average surface temperature, and $A_{0}$ is the amplitude of the seasonal surface temperature variation. Source: Applications in School Mathematics 1979 Yearbook. a. At what minimum depth $x$ is the amplitude of $u(x, t)$ at most $1^{\circ} \mathrm{C} ?$ b. Suppose we wish to construct a cellar to keep wine at a temperature between $14^{\circ} \mathrm{C}$ and $18^{\circ} \mathrm{C}$ . What minimum depth will accomplish this? C. At what minimum depth $x$ does the ground temperature model predict that it will be winter when it is summer at the surface, and vice versa? That is, when will the phase shift correspond to 1$/ 2$ year? d. Show that the ground temperature model satisfies the heal equation