Question 3.
It WAS remded in the analysisof molded synoptic data that the atmospheric pressure varies in time and spare in a certain Earth region in accordance with the following formula:
P(z.y,r)•-• Ih[l — sin Ur • cos 2sy • e-1,
where Pe 10; Pascals, r =0.ghrs, and x and y are dimensioning< %wallies ranging in the limits: —0.5 < r < 0.5 and —0.5 < y < 0.5.
(a) Find the prertue distribution within the inclicatal region after long time, say, after one day.
(b) Find the partial der-initial OP/Ox at the points (-1/2.0) and (0.0). What do those partial derivatives tell about the pressure behaviour in the vicitiity of indented points?
(c) Plot the nonnalised preagure P(z, 9,1)/ Po at the initial instant of time in two perpendicular crossanctkon: x = 1/4 and g = 0. Plot in the since frame. by hand or technology. using different colours.
(d) Find the partial &sant ive OP/Of ILK a function of y in the nornectiou x •• 1/4. Plot. by hand or tedmoloftv, this partial derivative against it for 1 = 0 and x = 1/4 and make a prediction regarding the pressure behaviour in the following few hours for this Esau-section.
Question 9.
After 3 clays a sample of radon-222 decayed to 58% of its initial amount.
(a) What is the half-life of radon-222 (period of decay to 1/2 of the initial amount)?
(b) How long would it take for the sample to decay to 10% of its initial amount?
Question 10.
Current in an electric circuit is described by the differential equation
dl
L— + RI E , dt
where L, R and E are positive constants. At the moment I = 0 there is no current: I = 0. Solve the differential equation to find 1(0 in terms of the constants and sketch the solution indicating coordinates that you think are important. What will be the practical value of the current after a very long time?
