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Ground Temperature Mathematical models of ground temperature variation usually involve Fourier series or other sophisticated methods. However, the elementary model ...

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



Answers

DATA ANALYSIS:
METEOROLOGY


The table shows the maximum daily high temperatures in Las Vegas $L$ and International Falls $I$ (in degrees Fahrenheit) for month $t$, with $t=1$ corresponding to January. (Source: National Climatic Data Center)

(a)
A model for the temperature in Las Vegas is given by

$L(t) =\ 80.60\ +\ 23.50\ cos(\frac{\pi t}{6} -\ 3.67)$.

Find a trigonometric model for International Falls.

(b)
Use a graphing utility to graph the data points and the model for the temperatures in Las Vegas. How well does the model fit the data?

(c)
Use a graphing utility to graph the data points and the model for the temperatures in International Falls. How well does the model fit the data?

(d)
Use the models to estimate the average maximum temperature in each city. Which term of the models did you use? Explain.

(e)
What is the period of each model? Are the periods what you expected? Explain.

(f )
Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.


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