(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Math 225: Calculus III Assignment 6 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] Signature: :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Instructions :[font = text; inactive; preserveAspect] This assignment contains problems on finding the extreme values of functions of several variables, including problems that use Lagrange multipliers, see Chapter 13 in Finney & Thomas or Chapter 3 in the Lecture Notes. Before you begin, you may want to review the notebooks MaxMin and AdvancedCalculusDemo in the Notebook folder or the back of the Lecture Notes. There are also many examples of how to do these problems in the Quizzes folder. Be sure to type in comments explaining what you are doing. Remember to uncheck Show In/Out Names in the File menu and close this group before printing. ;[s] 9:0,0;273,2;279,0;284,2;304,0;523,1;540,0;548,1;552,0;596,-1; 3:5,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect] *) < 0, x -> 0}, {y -> 0, x -> 0}, {y -> -0.131403921942119, x -> 0.0947105882913179}, {y -> 0.04878283932103644, x -> 0.06768257410184454}} ;[o] {{y -> 0, x -> 0}, {y -> 0, x -> 0}, {y -> -0.131404, x -> 0.0947106}, {y -> 0.0487828, x -> 0.0676826}} :[font = text; inactive; preserveAspect] All three are inside the rectangle. :[font = input; preserveAspect; startGroup] ii = f[x,y] /. cp :[font = output; output; inactive; preserveAspect; endGroup] {0, 0, 0.00143891589181421, 0.0001983137843518382} ;[o] {0, 0, 0.00143892, 0.000198314} :[font = text; inactive; preserveAspect] Now examine f[x,y] on the boundaries. Left side: ;[s] 3:0,0;12,1;18,0;49,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] left[x_] = f[x,-1/2] :[font = output; output; inactive; preserveAspect; endGroup] 1/16 - (3*x)/4 + x^2/2 + x^3 ;[o] 2 1 3 x x 3 -- - --- + -- + x 16 4 2 :[font = input; preserveAspect; startGroup] CriticalPoints[left[x]] //N :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -0.6937129433613965}, {x -> 0.3603796100280632}} ;[o] {{x -> -0.693713}, {x -> 0.36038}} :[font = input; preserveAspect; startGroup] ll = left[x] /. % :[font = output; output; inactive; preserveAspect; endGroup] {0.4895627463118869, -0.0960442277933685} ;[o] {0.489563, -0.0960442} :[font = text; inactive; preserveAspect] Right side: :[font = input; preserveAspect; startGroup] right[x_] = f[x,1] :[font = output; output; inactive; preserveAspect; endGroup] 1/4 - 3*x - x^2 + x^3 ;[o] 1 2 3 - - 3 x - x + x 4 :[font = input; preserveAspect; startGroup] CriticalPoints[right[x]] //N :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -0.7207592200561264}, {x -> 1.387425886722793}} ;[o] {{x -> -0.720759}, {x -> 1.38743}} :[font = input; preserveAspect; startGroup] rr = right[x] /. % :[font = output; output; inactive; preserveAspect; endGroup] {1.518353822346948, -3.166501970495096} ;[o] {1.51835, -3.1665} :[font = text; inactive; preserveAspect] Back side: :[font = input; preserveAspect; startGroup] back[y_] = f[-1,y] :[font = output; output; inactive; preserveAspect; endGroup] -1 - y + (13*y^2)/4 ;[o] 2 13 y -1 - y + ----- 4 :[font = input; preserveAspect; startGroup] CriticalPoints[back[y]] //N :[font = output; output; inactive; preserveAspect; endGroup] {{y -> 0.1538461538461539}} ;[o] {{y -> 0.153846}} :[font = input; preserveAspect; startGroup] bb = back[y] /. % :[font = output; output; inactive; preserveAspect; endGroup] {-1.076923076923077} ;[o] {-1.07692} :[font = text; inactive; preserveAspect] Front side: :[font = input; preserveAspect; startGroup] front[y_] = f[1,y] :[font = output; output; inactive; preserveAspect; endGroup] 1 - y - (11*y^2)/4 ;[o] 2 11 y 1 - y - ----- 4 :[font = input; preserveAspect; startGroup] CriticalPoints[front[y]] //N :[font = output; output; inactive; preserveAspect; endGroup] {{y -> -0.1818181818181818}} ;[o] {{y -> -0.181818}} :[font = input; preserveAspect; startGroup] ff = front[y] /. % :[font = output; output; inactive; preserveAspect; endGroup] {1.090909090909091} ;[o] {1.09091} :[font = text; inactive; preserveAspect] We must also examine the corners :[font = input; preserveAspect; startGroup] cc = f[x,y] /. {{x->-1,y->-1/2},{x->-1,y->1}, {x-> 1,y->-1/2},{x-> 1,y->1}} :[font = output; output; inactive; preserveAspect; endGroup] {5/16, 5/4, 13/16, -11/4} ;[o] 5 5 13 11 {--, -, --, -(--)} 16 4 16 4 :[font = text; inactive; preserveAspect] Now compare all these values (at critical points inside and on the boundaries, as well as the corner points) to find the absolute maximum and minimum: :[font = input; preserveAspect; startGroup] Max[ii,ll,rr,bb,ff,cc] :[font = output; output; inactive; preserveAspect; endGroup] 1.518353822346948 ;[o] 1.51835 :[font = text; inactive; preserveAspect] The maximum is 1.51835 and it occurs at {-0.720759,1}. ;[s] 6:0,0;15,1;22,0;40,1;53,0;54,1;55,-1; 2:3,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Min[ii,ll,rr,bb,ff,cc] :[font = output; output; inactive; preserveAspect; endGroup] -3.166501970495096 ;[o] -3.1665 :[font = text; inactive; preserveAspect] The minimum is -3.1665 and it occurs at {1.38742588,1}. The maximum and minimum are both on the right side of the rectangle. ;[s] 5:0,0;15,1;22,0;40,1;54,0;125,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; endGroup] Plot3D[f[x,y],{x,-1,1}, {y,-1/2,1}, ViewPoint->{2.988, 1.076, 1.168}] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 2 :[font = text; inactive; preserveAspect] Find the point(s) nearest the origin on the curve x^2 - x y + y^2 - 2 x + y == 5 by using Lagrange multipliers to find the minimum of the squared-distance function f[x_,y_] = x^2 + y^2 subject to the above constraint. ;[s] 5:0,0;50,1;83,0;167,1;187,0;221,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = input; preserveAspect; startGroup] lp = Lagrange[x^2 + y^2, x^2 - x y + y^2 - 2 x + y == 5] // Simplify :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -1, y -> -1 - 3^(1/2)}, {x -> -1, y -> -1 + 3^(1/2)}, {x -> (5 - 5^(1/2))/2, y -> -5^(1/2)}, {x -> (5 + 5^(1/2))/2, y -> 5^(1/2)}} ;[o] {{x -> -1, y -> -1 - Sqrt[3]}, {x -> -1, y -> -1 + Sqrt[3]}, 5 - Sqrt[5] {x -> -----------, y -> -Sqrt[5]}, 2 5 + Sqrt[5] {x -> -----------, y -> Sqrt[5]}} 2 :[font = text; inactive; preserveAspect] Evaluate the squared-distance function at these points: :[font = input; preserveAspect; startGroup] d = x^2+y^2 /. lp //Simplify :[font = output; output; inactive; preserveAspect; endGroup] {5 + 2*3^(1/2), 5 - 2*3^(1/2), (5*(5 - 5^(1/2)))/2, (5*(5 + 5^(1/2)))/2} ;[o] 5 (5 - Sqrt[5]) 5 (5 + Sqrt[5]) {5 + 2 Sqrt[3], 5 - 2 Sqrt[3], ---------------, ---------------} 2 2 :[font = input; preserveAspect; startGroup] N[d] :[font = output; output; inactive; preserveAspect; endGroup] {8.46410161513775, 1.535898384862246, 6.909830056250526, 18.09016994374947} ;[o] {8.4641, 1.5359, 6.90983, 18.0902} :[font = text; inactive; preserveAspect] The closest point is thus {-1,-1+Sqrt[3]}, at a distance of 5 - 2 Sqrt[3]. Note that {-1,-1+Sqrt[3]} is on the curve since ;[s] 7:0,0;26,1;41,0;60,1;73,0;85,1;100,0;123,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] x^2 - x y + y^2 - 2 x + y /. {x->-1, y->-1+Sqrt[3]} // Simplify :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 5 ;[o] 5 :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 3 :[font = text; inactive; preserveAspect] Find the extreme values of f[x_,y_] = x^2 + 3y^2 + 2y on the circular disk x^2 + y^2 <= 1. ;[s] 5:0,0;28,1;54,0;76,1;90,0;92,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = input; preserveAspect; startGroup] f[x_,y_] = x^2 + 3y^2 + 2y :[font = output; output; inactive; preserveAspect; endGroup] x^2 + 2*y + 3*y^2 ;[o] 2 2 x + 2 y + 3 y :[font = text; inactive; preserveAspect] Critical points inside the disk: :[font = input; preserveAspect; startGroup] cp = CriticalPoints[f[x,y]] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 0, y -> -1/3}} ;[o] 1 {{x -> 0, y -> -(-)}} 3 :[font = input; preserveAspect; startGroup] ii = f[x,y] /. cp :[font = output; output; inactive; preserveAspect; endGroup] {-1/3} ;[o] 1 {-(-)} 3 :[font = text; inactive; preserveAspect] Critical points on the boundary x^2 + y^2 == 1 can be found with Lagrange multipliers: ;[s] 3:0,0;32,1;47,0;87,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] lp = Lagrange[f[x,y], x^2 + y^2 == 1] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 0, y -> -1}, {x -> 0, y -> 1}, {x -> -3^(1/2)/2, y -> -1/2}, {x -> 3^(1/2)/2, y -> -1/2}} ;[o] -Sqrt[3] 1 {{x -> 0, y -> -1}, {x -> 0, y -> 1}, {x -> --------, y -> -(-)}, 2 2 Sqrt[3] 1 {x -> -------, y -> -(-)}} 2 2 :[font = input; preserveAspect; startGroup] bb = f[x,y] /. lp :[font = output; output; inactive; preserveAspect; endGroup] {1, 5, 1/2, 1/2} ;[o] 1 1 {1, 5, -, -} 2 2 :[font = input; preserveAspect; startGroup] Max[ii,bb] :[font = output; output; inactive; preserveAspect; endGroup] 5 ;[o] 5 :[font = input; preserveAspect; startGroup] Min[ii,bb] :[font = output; output; inactive; preserveAspect; endGroup] -1/3 ;[o] 1 -(-) 3 :[font = text; inactive; preserveAspect] The maximum is 5 and occurs at {0,1}. The minimum is -1/3 and occurs at {0,-1/3}. ;[s] 9:0,0;15,1;16,0;31,1;36,0;53,1;57,0;72,1;80,0;82,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] polar = {x,y,f[x,y]} /. {x->r Cos[t], y->r Sin[t]} //Simplify :[font = output; output; inactive; preserveAspect; endGroup] {r*Cos[t], r*Sin[t], r*(2*r - r*Cos[2*t] + 2*Sin[t])} ;[o] {r Cos[t], r Sin[t], r (2 r - r Cos[2 t] + 2 Sin[t])} :[font = input; preserveAspect; endGroup] ParametricPlot3D[polar, {r,0,1}, {t,0,2Pi}] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 4 :[font = text; inactive; preserveAspect] Suppose that the temperature T in degrees at the point {x,y,z} on the sphere x^2 + y^2 + z^2 == 1 is T[x_,y_,z_] = 10(1-z^2) + x^2 y^2. Locate where the highest and lowest temperatures on the sphere occur and what the temperatures are at those points. (Use Lagrange multipliers and be sure to ignore any complex numbers that arise.) ;[s] 9:0,0;29,1;30,0;55,1;62,0;77,1;97,0;101,1;134,0;333,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = input; preserveAspect; startGroup] T[x_,y_,z_] = 10(1-z^2) + x^2 y^2 :[font = output; output; inactive; preserveAspect; endGroup] x^2*y^2 + 10*(1 - z^2) ;[o] 2 2 2 x y + 10 (1 - z ) :[font = input; preserveAspect; startGroup] lp = Lagrange[T[x,y,z], x^2 + y^2 + z^2 == 1] :[font = output; output; inactive; preserveAspect; endGroup] {{z -> -1, x -> 0, y -> 0}, {z -> -1, x -> 0, y -> 0}, {z -> -1, x -> 0, y -> 0}, {z -> -1, x -> 0, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 1, y -> 0}, {z -> 0, x -> 1, y -> 0}, {z -> 0, x -> -2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> -2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> -2^(-1/2), y -> 2^(-1/2)}, {z -> 0, x -> -2^(-1/2), y -> 2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> 2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> 2^(-1/2)}, {z -> 1, x -> 0, y -> 0}, {z -> 1, x -> 0, y -> 0}, {z -> 1, x -> 0, y -> 0}, {z -> 1, x -> 0, y -> 0}, {z -> -21^(1/2), x -> -I*10^(1/2), y -> -I*10^(1/2)}, {z -> -21^(1/2), x -> -I*10^(1/2), y -> I*10^(1/2)}, {z -> -21^(1/2), x -> I*10^(1/2), y -> -I*10^(1/2)}, {z -> -21^(1/2), x -> I*10^(1/2), y -> I*10^(1/2)}, {z -> 21^(1/2), x -> -I*10^(1/2), y -> -I*10^(1/2)}, {z -> 21^(1/2), x -> -I*10^(1/2), y -> I*10^(1/2)}, {z -> 21^(1/2), x -> I*10^(1/2), y -> -I*10^(1/2)}, {z -> 21^(1/2), x -> I*10^(1/2), y -> I*10^(1/2)}} ;[o] {{z -> -1, x -> 0, y -> 0}, {z -> -1, x -> 0, y -> 0}, {z -> -1, x -> 0, y -> 0}, {z -> -1, x -> 0, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 1, y -> 0}, {z -> 0, x -> 1, y -> 0}, 1 1 {z -> 0, x -> -(-------), y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -(-------), y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -(-------), y -> -------}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -(-------), y -> -------}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -------}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -------}, Sqrt[2] Sqrt[2] {z -> 1, x -> 0, y -> 0}, {z -> 1, x -> 0, y -> 0}, {z -> 1, x -> 0, y -> 0}, {z -> 1, x -> 0, y -> 0}, {z -> -Sqrt[21], x -> -I Sqrt[10], y -> -I Sqrt[10]}, {z -> -Sqrt[21], x -> -I Sqrt[10], y -> I Sqrt[10]}, {z -> -Sqrt[21], x -> I Sqrt[10], y -> -I Sqrt[10]}, {z -> -Sqrt[21], x -> I Sqrt[10], y -> I Sqrt[10]}, {z -> Sqrt[21], x -> -I Sqrt[10], y -> -I Sqrt[10]}, {z -> Sqrt[21], x -> -I Sqrt[10], y -> I Sqrt[10]}, {z -> Sqrt[21], x -> I Sqrt[10], y -> -I Sqrt[10]}, {z -> Sqrt[21], x -> I Sqrt[10], y -> I Sqrt[10]}} :[font = text; inactive; preserveAspect] To get rid of duplicates: :[font = input; preserveAspect; startGroup] lp = Union[lp] :[font = output; output; inactive; preserveAspect; endGroup] {{z -> -1, x -> 0, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 1, y -> 0}, {z -> 0, x -> -2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> -2^(-1/2), y -> 2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> 2^(-1/2)}, {z -> 1, x -> 0, y -> 0}, {z -> -21^(1/2), x -> -I*10^(1/2), y -> -I*10^(1/2)}, {z -> -21^(1/2), x -> -I*10^(1/2), y -> I*10^(1/2)}, {z -> -21^(1/2), x -> I*10^(1/2), y -> -I*10^(1/2)}, {z -> -21^(1/2), x -> I*10^(1/2), y -> I*10^(1/2)}, {z -> 21^(1/2), x -> -I*10^(1/2), y -> -I*10^(1/2)}, {z -> 21^(1/2), x -> -I*10^(1/2), y -> I*10^(1/2)}, {z -> 21^(1/2), x -> I*10^(1/2), y -> -I*10^(1/2)}, {z -> 21^(1/2), x -> I*10^(1/2), y -> I*10^(1/2)}} ;[o] {{z -> -1, x -> 0, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 1, y -> 0}, 1 1 {z -> 0, x -> -(-------), y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -(-------), y -> -------}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -------}, Sqrt[2] Sqrt[2] {z -> 1, x -> 0, y -> 0}, {z -> -Sqrt[21], x -> -I Sqrt[10], y -> -I Sqrt[10]}, {z -> -Sqrt[21], x -> -I Sqrt[10], y -> I Sqrt[10]}, {z -> -Sqrt[21], x -> I Sqrt[10], y -> -I Sqrt[10]}, {z -> -Sqrt[21], x -> I Sqrt[10], y -> I Sqrt[10]}, {z -> Sqrt[21], x -> -I Sqrt[10], y -> -I Sqrt[10]}, {z -> Sqrt[21], x -> -I Sqrt[10], y -> I Sqrt[10]}, {z -> Sqrt[21], x -> I Sqrt[10], y -> -I Sqrt[10]}, {z -> Sqrt[21], x -> I Sqrt[10], y -> I Sqrt[10]}} :[font = text; inactive; preserveAspect] Drop the last 8 solutions since they are complex: :[font = input; preserveAspect; startGroup] lpr = Drop[lp , -8] :[font = output; output; inactive; preserveAspect; endGroup] {{z -> -1, x -> 0, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 1, y -> 0}, {z -> 0, x -> -2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> -2^(-1/2), y -> 2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> -2^(-1/2)}, {z -> 0, x -> 2^(-1/2), y -> 2^(-1/2)}, {z -> 1, x -> 0, y -> 0}} ;[o] {{z -> -1, x -> 0, y -> 0}, {z -> 0, x -> -1, y -> 0}, {z -> 0, x -> 0, y -> -1}, {z -> 0, x -> 0, y -> 1}, {z -> 0, x -> 1, y -> 0}, 1 1 {z -> 0, x -> -(-------), y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -(-------), y -> -------}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -(-------)}, Sqrt[2] Sqrt[2] 1 1 {z -> 0, x -> -------, y -> -------}, Sqrt[2] Sqrt[2] {z -> 1, x -> 0, y -> 0}} :[font = input; preserveAspect; startGroup] T[x,y,z] /. lpr :[font = output; output; inactive; preserveAspect; endGroup] {0, 10, 10, 10, 10, 41/4, 41/4, 41/4, 41/4, 0} ;[o] 41 41 41 41 {0, 10, 10, 10, 10, --, --, --, --, 0} 4 4 4 4 :[font = text; inactive; preserveAspect; endGroup] The maximum temperature is 41/4 == 10.25 which occurs at the four points {± 1/Sqrt[2], ±1/Sqrt[2], 0}. The minimum is 0 which occurs at the two points {0,0,±1}. ;[s] 9:0,0;27,1;40,0;73,1;101,0;118,1;119,0;151,1;159,0;161,-1; 2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Problem 5 :[font = text; inactive; preserveAspect] Your firm has been asked to design a storage tank for liquid petroleum gas. The customer's specifications call for a cylindrical tank that holds 8000 m^3 of gas. The customer also wants to use the smallest amount of material possible in building the tank. What radius and height do you recommend for the tank? :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect] Solution :[font = text; inactive; preserveAspect] The cost of material is proportional to the surface area so we must minimize (side and two ends): :[font = input; preserveAspect; startGroup] s[r_,h_] = 2Pi r h + 2 Pi r^2 :[font = output; output; inactive; preserveAspect; endGroup] 2*h*Pi*r + 2*Pi*r^2 ;[o] 2 2 h Pi r + 2 Pi r :[font = text; inactive; preserveAspect] Subject to the constraint on the volume: :[font = input; preserveAspect] Pi r^2 h == 8000 :[font = input; preserveAspect; startGroup] lp = Lagrange[s[r,h], Pi r^2 h == 8000, {r,h}] :[font = output; output; inactive; preserveAspect; endGroup] {{h -> 20*(4/Pi)^(1/3), r -> 10*(4/Pi)^(1/3)}, {h -> -20*(-1)^(1/3)*(4/Pi)^(1/3), r -> -10*(-1)^(1/3)*(4/Pi)^(1/3)}, {h -> 20*(-1)^(2/3)*(4/Pi)^(1/3), r -> 10*(-1)^(2/3)*(4/Pi)^(1/3)}} ;[o] 4 1/3 4 1/3 {{h -> 20 (--) , r -> 10 (--) }, Pi Pi 1/3 4 1/3 1/3 4 1/3 {h -> -20 (-1) (--) , r -> -10 (-1) (--) }, Pi Pi 2/3 4 1/3 2/3 4 1/3 {h -> 20 (-1) (--) , r -> 10 (-1) (--) }} Pi Pi :[font = input; preserveAspect; startGroup] N[lp] :[font = output; output; inactive; preserveAspect; endGroup] {{h -> 21.67704280557156, r -> 10.83852140278578}, {h -> -10.83852140278578 - 18.77286974854768*I, r -> -5.419260701392892 - 9.38643487427384*I}, {h -> -10.83852140278578 + 18.77286974854768*I, r -> -5.419260701392891 + 9.38643487427384*I}} ;[o] {{h -> 21.677, r -> 10.8385}, {h -> -10.8385 - 18.7729 I, r -> -5.41926 - 9.38643 I}, {h -> -10.8385 + 18.7729 I, r -> -5.41926 + 9.38643 I}} :[font = input; preserveAspect; startGroup] s[r,h] /. lp :[font = output; output; inactive; preserveAspect; endGroup] {1200*(2*Pi)^(1/3), 1200*(-1)^(2/3)*(2*Pi)^(1/3), -1200*(-1)^(1/3)*(2*Pi)^(1/3)} ;[o] 1/3 2/3 1/3 {1200 (2 Pi) , 1200 (-1) (2 Pi) , 1/3 1/3 -1200 (-1) (2 Pi) } :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] {2214.324178372834, -1107.162089186417 + 1917.660990684979*I, -1107.162089186417 - 1917.660990684979*I} ;[o] {2214.32, -1107.16 + 1917.66 I, -1107.16 - 1917.66 I} :[font = text; inactive; preserveAspect; endGroup] The minimum surface area is 1200 (2Pi)^(1/3) == 2214.32 and occurs when h == 20(4/Pi)^(1/3) == 21.677, r == 10(4/Pi)^(1/3) == 10.8385 ;[s] 6:0,0;28,1;55,0;72,1;101,0;102,1;134,-1; 2:3,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; ^*)