= 2. You are mounting an expedition to reach the other side of a volcano, and you wish to determine the path that will minimize the distance traveled. To perform this calculation you decide to use cylindrical coordinates (2,0, z) and you model the volcano as the conical surface z = 1 - p. [The cylindrical coordinates are defined by x = p cos 0, y = psin o, and z = z.] You describe the path by the function plo) and let the angle o range through the interval - 505. The starting point of the expedition is (p=1,0 = -2, z = 0) and the end point is (p = 1,0 = +, z = 0). You wish to find the path plo) that minimizes the total distance traveled from this side of the volcano to the other side. = (a) Prove that the functional that must be minimized is slp] = L V2cp2 + pp d where p = dp/do. (b) Find the differential equation that the minimal path must satisfy. (c) Show that (272) P(0) I COS COS = is a solution to this differential equation, and that it satisfies the bound- ary conditions; conclude that this must be the minimal path. Produce a plot of z=1-p as a function of o. (d) Calculate the minimum distance Smin. Compare this with the distance that would be traveled if the path were instead chosen to be pl) = = 1. = 2. You are mounting an expedition to reach the other side of a volcano, and you wish to determine the path that will minimize the distance traveled. To perform this calculation you decide to use cylindrical coordinates (2,0, z) and you model the volcano as the conical surface z = 1 - p. [The cylindrical coordinates are defined by x = p cos 0, y = psin o, and z = z.] You describe the path by the function plo) and let the angle o range through the interval - 505. The starting point of the expedition is (p=1,0 = -2, z = 0) and the end point is (p = 1,0 = +, z = 0). You wish to find the path plo) that minimizes the total distance traveled from this side of the volcano to the other side. = (a) Prove that the functional that must be minimized is slp] = L V2cp2 + pp d where p = dp/do. (b) Find the differential equation that the minimal path must satisfy. (c) Show that (272) P(0) I COS COS = is a solution to this differential equation, and that it satisfies the bound- ary conditions; conclude that this must be the minimal path. Produce a plot of z=1-p as a function of o. (d) Calculate the minimum distance Smin. Compare this with the distance that would be traveled if the path were instead chosen to be pl) = = 1
