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" }}{PARA 0 "" 0 "" {TEXT -1 82 "Eac h has a square base and will hold 1000 cubic feet. It orders a pre-c ut sheet " }}{PARA 0 "" 0 "" {TEXT -1 81 "for the bottom of the box an d another that it bends three times to form the four " }}{PARA 0 "" 0 "" {TEXT -1 79 "sides. (There is no top.) It must then weld the seam s (one vertical and four " }}{PARA 0 "" 0 "" {TEXT -1 80 "horizontal). Records indicate that welding costs $2.10 per foot includig labor " }}{PARA 0 "" 0 "" {TEXT -1 62 "and materials. The sheet metal costs $ 1.85 per square foot. " }}{PARA 0 "" 0 "" {TEXT -1 69 "a) What are t he dimensions of the box that costs the least to build?" }}{PARA 0 "" 0 "" {TEXT -1 54 "b) What is the cost of the box described in Part (a )?" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 260 10 "Solution: " }}{SECT 0 {PARA 4 "" 0 "" {TEXT 268 39 "Translate the information i nto algebra." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 283 1 "h" } {TEXT -1 67 " be the height of the box and b the length (and width) of the base." }}{PARA 0 "" 0 "" {TEXT -1 58 " Total cost = cost of metal + welding cost" }}{PARA 0 "" 0 "" {TEXT -1 17 "Area o f bottom = " }{XPPEDIT 18 0 "b^2;" "6#*$%\"bG\"\"#" }{TEXT -1 31 " sq . ft., Area of 4 sides = 4" }{TEXT 270 2 "bh" }{TEXT -1 11 " sq. ft. \+ " }}{PARA 0 "" 0 "" {TEXT -1 25 "Perimeter of the base = 4" }{TEXT 269 1 "b" }{TEXT -1 27 " ft. height of the seam = " }{TEXT 271 1 "h" }{TEXT -1 4 " ft." }}{PARA 0 "" 0 "" {TEXT -1 23 "Cost of metal = 1.85 ( " }{XPPEDIT 18 0 "b^2;" "6#*$%\"bG\"\"#" }{TEXT -1 4 " + 4" }{TEXT 273 2 "bh" }{TEXT -1 11 ") dollars. " }}{PARA 0 "" 0 "" {TEXT -1 21 "W elding cost = 2.10(4" }{TEXT 272 1 "b" }{TEXT -1 3 " + " }{TEXT 274 1 "h" }{TEXT -1 10 ") dollars." }}{PARA 0 "" 0 "" {TEXT -1 37 " \+ Total cost = 1.85(" }{XPPEDIT 18 0 "b^2;" "6#*$%\"bG\"\"#" } {TEXT -1 3 "+ 4" }{TEXT 277 2 "bh" }{TEXT -1 10 ") + 2.10(4" }{TEXT 278 1 "b" }{TEXT -1 3 " + " }{TEXT 279 1 "h" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 21 " Volume of the box = " }{TEXT 280 1 "h" } {XPPEDIT 18 0 "b^2;" "6#*$%\"bG\"\"#" }{TEXT -1 8 " = 1000 " }}{PARA 0 "" 0 "" {TEXT -1 47 " This equation allows us to express \+ " }{TEXT 281 1 "h" }{TEXT -1 13 " in terms of " }{TEXT 276 1 "b" } {TEXT -1 3 ": " }{TEXT 282 1 "h" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "10 00/(b^2);" "6#*&\"%+5\"\"\"*$%\"bG\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 35 " Therefore, the total cost = 1.85(" }{XPPEDIT 18 0 "b^2;" "6#* $%\"bG\"\"#" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "4000*b/(b^2);" "6#*(\" %+S\"\"\"%\"bGF%*$F&\"\"#!\"\"" }{TEXT -1 10 ") + 2.10(4" }{TEXT 275 1 "b" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1000/(b^2);" "6#*&\"%+5\"\"\"* $%\"bG\"\"#!\"\"" }{TEXT -1 4 ") " }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ = 1.85(" }{XPPEDIT 18 0 "b^2;" "6 #*$%\"bG\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "4000/b;" "6#*&\"%+S\" \"\"%\"bG!\"\"" }{TEXT -1 14 ") + 2.10(4b + " }{XPPEDIT 18 0 "1000/(b^ 2)" "6#*&\"%+5\"\"\"*$%\"bG\"\"#!\"\"" }{TEXT -1 3 ") " }}{PARA 0 "" 0 "" {TEXT -1 55 " We now define the cost function in Maple. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "cost := 1.85*(b^2 + 4000/b) + 2.10*(4*b + 1000/b^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%costG,**&$\"$&=!\"#\"\"\")%\"bG\"\"#F*F**&$\"'++uF)F*F,!\"\"F**&$ \"$S)F)F*F,F*F**&$\"'++@F)F*F,F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dcost := diff(cost, b);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dcostG,**&$\"$q$!\"#\"\"\"%\"bGF*F**&$\"'++uF)F*F+F) !\"\"$\"$S)F)F**&$\"'++UF)F*F+!\"$F/" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 284 11 "a) Method 1" }{TEXT -1 3 " - " }{TEXT 285 38 "Graph and find the minimum by zooming." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots); \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Z%(animateG%*animate3dG%-animat ecurveG%&arrowG%-changecoordsG%,complexplotG%.complexplot3dG%*conforma lG%,conformal3dG%,contourplotG%.contourplot3dG%*coordplotG%,coordplot3 dG%-cylinderplotG%,densityplotG%(displayG%*display3dG%*fieldplotG%,fie ldplot3dG%)gradplotG%+gradplot3dG%,graphplot3dG%-implicitplotG%/implic itplot3dG%(inequalG%,interactiveG%-listcontplotG%/listcontplot3dG%0lis tdensityplotG%)listplotG%+listplot3dG%+loglogplotG%(logplotG%+matrixpl otG%(odeplotG%'paretoG%,plotcompareG%*pointplotG%,pointplot3dG%*polarp lotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.polyhedraplo tG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoptions3dG%+spa cecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplotG%+textpl ot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := b \+ -> diff(cost, b);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"bG 6\"6$%)operatorG%&arrowGF(-%%diffG6$%%costG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(b), b=5..20);\n" }}{PARA 13 "" 1 " " {GLPLOT2D 212 154 154 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"\"&\"\"!$!3- ,+++++FI!#:7$$\"3o***\\ilyM;&!#<$!39&=0]rtb!GF-7$$\"3E++]7t&pK&F1$!3o \"\\47SZXg#F-7$$\"3o*\\(=<#*>paF1$!31/)*H/CGWCF-7$$\"3****\\(=7T9h&F1$ !3#[]**[LchH#F-7$$\"3<++v=HPJfF1$!3.\"p(=f$37+#F-7$$\"3;++DJaU`iF1$!3g j_j%*yp[W%*G`\"F-7$$\"3G+](=276(oF1$!3-Q) =bnI'e8F-7$$\"3S+](o**3)yrF1$!3mSCWP#=)*>\"F-7$$\"3[+](ofHq\\(F1$!3%eg c'y?)[0\"F-7$$\"3.+]Pf'HU\"yF1$!39ns9'RZxE*!#;7$$\"3k****\\7*309)F1$!3 #oACQ*GP$4)F`o7$$\"3:++Dce*yU)F1$!3E>umJ\"*\\hrF`o7$$\"3g++]([D9v)F1$! 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HP%3(fQKC\"F-7$$\"3\"\\i.tK$)y?\"F*$!3WsK#)f#*\\)=\"F-7$$\"3(\\(3zMu)y ?\"F*$!3,,cmg[PO6F-7$$\"3e;H_?<*y?\"F*$!3!3Nb(G*))>3\"F-7$$\"3b\"zihl& *y?\"F*$!3C$)f'\\.Y?.\"F-7$$\"3Q$3#G,***y?\"F*$!3O+lqJFw\"y*!#?7$$\"3I ezw5V!z?\"F*$!32Xqte;@A#*F`s7$$\"3-v$Q#\\\"3z?\"F*$!3D*)>q;W7N()F`s7$$ \"3H$e\"*[H7z?\"F*$!3&QeRsig!4#)F`s7$$\"3#***pvxl\"z?\"F*$!3CjG_v$*elw F`s7$$\"3(*\\_qn2#z?\"F*$!39y>ZLG\"R8(F`s7$$\"3,Dcp@[#z?\"F*$!3^O8O`A \\>mF`s7$$\"3\"*\\2'HKHz?\"F*$!3Wm*R&)H?$[gF`s7$$\"3nmwanL$z?\"F*$!3') p&)>O35NbF`s7$$\"3'**\\2goPz?\"F*$!3]Xy]NO8()\\F`s7$$\"3QeR<*fTz?\"F*$ !3/*G!=#e(f!\\%F`s7$$\"3%**\\)Hxe%z?\"F*$!3!p405_cx%RF`s7$$\"3u\"H!o-* \\z?\"F*$!3&Gag(*>*)pV$F`s7$$\"3#\\7k.6az?\"F*$!3ko2uHT4.HF`s7$$\"3pmT 9C#ez?\"F*$!3#4gAk,=6Q#F`s7$$\"3+D1*3`iz?\"F*$!39j*yl`hY$=F`s7$$\"3UL$ *zym'z?\"F*$!3,xHo)*HO38F`s7$$\"3F$3N1#4(z?\"F*$!3Vl=Nzy],x!#@7$$\"3i \"HYt7vz?\"F*$!3#)yfn(*H*RO#Ffx7$$\"3&***p(G**yz?\"F*$\"3^_cVbKfSDFfx7 $$\"3h;9@BM)z?\"F*$\"3/$HpTmg<;)Ffx7$$\"3UL`v&Q()z?\"F*$\"3nVCP(=O*=8F `s7$$\"33DOl5;*z?\"F*$\"3Wt!RGRz\\&=F`s7$$\"3)\\P?Wl&*z?\"F*$\"3mu#QXK P!oBF`s7$$\"3,++++++37F*$\"3!=ZR>)RQ>HF`s-%'COLOURG6&%$RGBG$\"#5!\"\"$ \"\"!Fb[lFa[l-%+AXESLABELSG6$Q\"b6\"Q!Fg[l-%%VIEWG6$;$\"&y?\"!\"$$\"%3 7!\"#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 44.000000 46.000000 0 0 "Curve 1" }}}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 286 48 "a) Method 2 -Compute minimum by differentiation." }}{PARA 0 "" 0 " " {TEXT -1 97 "Solving dcost = 0 for b we get similar answers as seen \+ below. We only compute b at this step. " }}{PARA 0 "" 0 "" {TEXT -1 35 "We use the 'fsolve' Maple command. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "fsolve(dcost = 0, b, 10..20);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+\"*p(z?\"!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 287 11 "Conclusion:" }{TEXT 289 1 " " }{TEXT -1 42 "Bot h methods give rise to the same answer:" }{TEXT 288 3 " " }{TEXT -1 15 "b = 12.07977 ft" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 290 25 "Compute h and dimensions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h:= 100 0/(12.07977)^2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"+q0.`o! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 96 "The dimensions of the smallest bo x with volume = 1000 cu. ft. are b = 12.0797 ft, h = 6.8531 ft." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 291 28 "b) Compute the mi nimum cost." }}{PARA 0 "" 0 "" {TEXT -1 42 "We substitute b = 12.0798 \+ in the function " }{XPPEDIT 18 0 "cost := 1.85*b^2+7400.00*1/b+8.40*b+ 2100.00*1/(b^2);" "6#>%%costG,**&-%&FloatG6$\"$&=!\"#\"\"\"*$%\"bG\"\" #F,F,*(-F(6$\"'++uF+F,F,F,F.!\"\"F,*&-F(6$\"$S)F+F,F.F,F,*(-F(6$\"'++@ F+F,F,F,*$F.F/F4F," }}{PARA 0 "" 0 "" {TEXT 292 1 " " }{TEXT 293 29 "W e use 'subs' Maple command. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs (b = 12.0797, cost);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +uV4%)**!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}}}{PARA 3 "" 0 "" {TEXT -1 44 "____________________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 256 8 "Problem:" }}{PARA 0 "" 0 "" {TEXT -1 80 " A company has a c ontract to build several metal boxes with lids, which are " }}{PARA 0 "" 0 "" {TEXT -1 79 "not welded to the box. Construct a box with a t op 4 inches more on each side " }}{PARA 0 "" 0 "" {TEXT -1 79 "than th e base. Each has a square base and will hold 1000 cubic feet. The li d " }}{PARA 0 "" 0 "" {TEXT -1 76 "for each box is a square 4 inches m ore than the base. The company orders a " }}{PARA 0 "" 0 "" {TEXT -1 81 "pre-cut sheet for the bottom of the box and another that it bends \+ three times to " }}{PARA 0 "" 0 "" {TEXT -1 75 "form the four sides. \+ It must then weld the seams (one vertical and four " }}{PARA 0 "" 0 "" {TEXT -1 80 "horizontal). Records indicate that welding costs $2.1 0 per foot includig labor " }}{PARA 0 "" 0 "" {TEXT -1 62 "and materia ls. The sheet metal costs $1.85 per square foot. " }}{PARA 0 "" 0 " " {TEXT -1 69 "a) What are the dimensions of the box that costs the l east to build?" }}{PARA 0 "" 0 "" {TEXT -1 54 "b) What is the cost of the box described in Part (a)?" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}} {PARA 3 "" 0 "" {TEXT 266 64 "________________________________________ ________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 99 "MSIP Grant #P120A80089-98: \"Three Urban Calculus Reform programs: Adopting the Best\" 1998-2001 " }}{PARA 259 "" 0 "" {TEXT 261 1 " " }}{PARA 260 "" 0 "" {TEXT 262 0 "" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{MARK "9 1 15 4 1 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }