(10/8/08)Math 10C. Lecture Examples.Section 12.5. Functions of three variables†Example 1 The frustum of a right circular cone with base of radius R (meters), top ofradius r (meters), and height h (meters) (Figure 1) has volumeV(R, r, h) =13π(R2+ rR + r2)h.(a) What are the domain and range of V?(b) What does V(6, 4, 9) represent and what is its value if lengths aremeasured in meters?(c) What is the geomet ric interpretation of V(R, 0, h)?FIGURE 1Answer: (a) The domain of V consists of all points (R, r, h) in three-dimensional Rrh-space such thatR ≥ 0, r ≥ 0, h ≥ 0. • Its range is the interval [0, ∞) on a V -axis.(b) V (9, 6, 4) is the volume of a frustum of a right circular cone with base of radius 9 meters, t o p of radius 6 meters,and 4 meters high. • V (9, 6, 4) = 228π cubic meters(c) V (R, 0, h) =13πR2h is the volume of a right circular cone whose base has radius R and whose height is h.(Figure A1)Figure A1†Lecture notes to accompany Section 12.5 of Calculu s by Hughes-Hallett et al.1Math 10C. Lecture Examples. (10/8/08) Section 12.5, p. 2Example 2 Describe t he level surfaces of f (x, y, z) = x2+ y2+ z2.Answer: For c = 0, the level surface f = c is the origin. • For c > 0 it is the surface of the sphere of radius√ccentered at the origin in Figure A2. • For c < 0 it is empty.Figure A2Example 3 Describe t he level surfaces of g(x, y, z ) = x2+ y2.Answer: The level s urface g = c is the z-axis if c = 0 , is the cylinder of radius√c with the z-axis as axis inFigure A3 if c > 0, and is empty if c < 0.Figure A3Section 12.5, p. 3 Math 10C. Lecture Examples. (10/8/08)Example 4 The level surface h(x, y, z) = 1 of h(x, y, z ) =19x2+116y2+ z2in Figure 2 iscalled an ellipsoid because all of its cross sections are ellipses. Find equationsfor and draw its cross sections ( a) in the xz-plane, (b) in t he yz-plane, and(c) in the xy-plane.A level surface ofh(x, y, z) =19x2+116y2+ z2FIGURE 2Answer: (a) The cross section19x2+ z2= 1 in the xz-plan e is the ellipse with x-intercepts x = ±3 and z-intercepts z = ±1 in Figure A4a.(b) The cross section116y2+ z2= 1 in the yz-plane is the ellipse with y-intercepts y = ±4 and z - interceptsz = ±1 in Figure A4b .(c) The the cross section19x2+116y2= 1 in the xy-plane is the ellipse with x-intercepts x = ±3 and y-interceptsy = ±4 in Figure A4c.x3z219x2+ z2= 1yz2116y2+ z2= 14x2y219x2+116y2= 1Figure A4a Figure A4b Figure A4cExample 5 Describe the level surfaces h = c of the function of Example 4 (a) withc > 1 and (b) with 0 < c < 1.Answer: (a) The level surfaces h = c with c > 1 are geometrically similar ellipsoids outside the ellipsoid h = 1of Figure 2. (b) The level surfaces h = c with 0 < c < 1 are geometrically similar ellipsoids inside the ellipsoidh = 1 of Figure 2.Math 10C. Lecture Examples. (10/8/08) Section 12.5, p. 4Example 6 Figure 3 is a cross-sectional view of level surfaces of the Van Allen belts ofcosmic radiation that surround the earth. (To visualize the level surfacesimagine that the curves as drawn are rotated around the north-south axisof the earth.) The radiation is measured in counts per second and the scaleshows the distance in earth radii (≈ 4000 miles). At approximately whatdistances from the equator is the radiation the greatest?FIGURE 3Answer: The radiation is greatest at approximately 0.5 earth radii and at approximately 2.5 earth radii from theequator.Example 7 On July 21, 2006, Ford Motor stock sold for $6.19 pe r share, HewlettPackard stock sold for $31.80 per share, and Motorola stock sold for $20 .60per share. Give a formula for the cost C(x1, x2, x3) on that day of x1sharesof Ford Motor stock, and x2shares of Hewlett Packard stock. What is thedomain of this function?Answer: C(x1, x2, x3) = 6.19x1+ 31.80x2+ 20 .60x3dollars. • The domain is the set of po ints’ (x1, x2, x3),such that x1, x2, and x3are all n onnegative integers.Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 14.7: Example 1‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the