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12 128 64 0 1 0 0 0 0 0 1 3 0 0 }0 0 0 -1 -1 -1 3 6 0 0 0 0 0 0 }{PSTYLE "subprobl em" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "diagram" -1 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "dblnorm.mws" -1 295 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Item" 0 296 1 {CSTYLE "" -1 -1 "Lucida Sans" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 3 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 286 "" 0 "" {TEXT -1 32 "Setting Up and Solving \+ Problems " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "What is a problem?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 546 " For our purposes, a problem is anything that can be formulated as a question whose answer involves s ome mathematics. The main use of mathematics is to solve problems, an d the best way to learn mathematics is to solve problems. This idea o f a problem includes the 'skill' exercises that are always at the end \+ of the sections in mathematics textbooks. For example, early in begin ning algebra there is a section on solving linear equations, and at th e end of that section there is a set of exercises consisting of lots o f problems like 'solve " }{XPPEDIT 18 0 "4x+3 = 2x -6" "/,&*&\"\"%\" \"\"%\"xGF&F&\"\"$F&,&*&\"\"#F&F'F&F&\"\"'!\"\"" }{TEXT -1 9 " for x' ." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 556 "It also includes 'word problems' which can be solved using the methods m astered in the skill exercises. The word problems often are stated at the beginning of the section as motivation for the methods which are \+ developed in the section. Problems usually arise in a context. Once \+ the context is well understood, the problem can be formulated or posed and a method of solution worked out. From this solution, other probl ems may arise which require solving. We want to consider this problem identification and formulation as part of problem solving also." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 237 "The pro cess of solving a problem is an active process, but can get bogged dow n for lack of knowing what to do next. So it is helpful to have a lis t of things to do. Here is one list of steps to carry out when you ar e solving a problem." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Setup - - Solve -- Interpret" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 271 18 "SETUP the problem." }{TEXT -1 30 " This involves several step s." }}{PARA 15 "" 0 "" {TEXT -1 104 " Pose/Read the problem carefull y, getting straight the meaning of all the terms used in the statement ." }}{PARA 15 "" 0 "" {TEXT -1 107 "Draw a picture or diagram. This i s a good way to focus thoughts, and gives you a place to put your labe ls." }}{PARA 15 "" 0 "" {TEXT -1 423 "Label or list the dimensions (or variables) important to the problem. Among these are the given dimen sions, that is, the dimensions whose values are stated in the problem, the requested dimensions, that is, the dimensions whose values are re quested in theproblem, and the intermediate dimensions, that is, the d imensions which arise in the process of trying to determine the reques ted dimensions from the given dimensions." }}{PARA 15 "" 0 "" {TEXT -1 62 " List or derive the equations relating the labeled dimensions. " }}{PARA 0 "" 0 "" {TEXT 272 20 "SOLVE the equations " }{TEXT -1 305 "we have set up for the required dimensions in terms of the given dime nsions. This is one of places where Maple comes in very handy. It is easy to get bogged down in the calculations so that you lose all in terest in solving the problem. This is less likely to happen if you ha ve Maple at your disposal." }}{PARA 0 "" 0 "" {TEXT 273 32 "INTERPRET \+ the obtained solutions" }{TEXT -1 130 ". What are the realistic solut ions given the context of the problem? Which should be ignored? Have all solutions been obtained?" }}{PARA 0 "" 0 "" {TEXT -1 289 " Where \+ does Maple come in handy in this process? In the Solve phase, mostly. The actual setting up of the problem as a mathematical problem has t o be done by you. Regard Maple as a tool to carry out and record the \+ solution you imagine. A very important word in the Maple vocabulary i s " }{TEXT 267 5 "solve" }{TEXT -1 80 ". This word is used to solve a system of one or more equations which come up." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "A Swimming Poo l Problem:" }}{EXCHG {PARA 289 "" 0 "" {TEXT -1 129 " Problem: A swimm ing pool is three times as long as it is wide. It is also 40 feet lon ger than it is wide. Find its dimensions." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 258 8 "Solution" }{TEXT -1 268 ": Le t l and w be the length and width of the pool. Then the first statemen t of the problem translates to the equation l = 3w , and the second \+ statement to l = w + 40 . We need to solve these two equations simu ltaneously for l and w. First, set up the equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " eq1 := l = \+ 3*w; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/%\"lG,$%\"wG\"\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " eq2 := l = w + 40 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/%\"lG,&%\"wG\"\"\"\"#SF) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "Then solve the system for l and w. We can do this \+ by subtracting eq2 from eq1 and solving for w, getting w = 20 , then substituting that value for w into eq1 getting l = 60 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " eq3 := eq1 - eq2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/\"\"!,&%\" wG\"\"#!#S\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " eq 4 := lhs(eq3) - 2*w = rhs(eq3) - 2*w; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq4G/,$%\"wG!\"#!#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " eq5 := -(1/2)*eq4; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq5G/%\"wG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " eq6 := subs(eq5,eq1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq6G/%\"lG\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " solution := \{eq5,eq6\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)solutionG<$/%\"wG\"#?/%\"lG\"#g" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "solve" {TEXT -1 9 "The word " }{TEXT 267 5 "solve" } {TEXT -1 45 " carries out this algorithm automatically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " solve( \{eq1,eq2\},\{l,w\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"wG\"#?/ %\"lG\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The pool \+ is 20 feet wide and 60 feet long. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 226 "The word solve is a very good to know, \+ but it is not infallible. It uses some methods of solving equations w hich you know and some which you probably don't know. Don't give up j ust because solve doesn't give you a solution. " }}}}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 33 "Four methods of solving equations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 " So it is important to also be able to solve eq uations by various means. Here are four." }}{PARA 0 "" 0 "" {TEXT 257 16 "Guess and check." }{TEXT -1 231 " This method involves guest imating a solution somehow and checking your accuracy somehow. For e xample, suppose you had guestimated that the dimensions of the pool ar e about 25 by 55 feet and wanted to check that. Use the word " } {TEXT 267 4 "subs" }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " subs(\{w=25,l=75\},[eq1,eq 2]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "This line says to substitute 25 and 75 for w and l in eq1 and eq2." }}{PARA 0 "" 0 "" {TEXT -1 583 "( Notice the use of braces and brackets here. Braces are used to enclo se the members of a set and brackets are used to enclose the members o f a list. In a set, the order is not important and repetitions are no t counted; in a list, the order is important and repetitions are count ed.) As we can clearly see, our guestimate is off. We have satisfied the first equation, but not the second. If we decrease the value of \+ w by 1, then we have to decrease the value of l by 3 in order to conti nue to satisfy the first equation. Will we come closer to satisfying \+ the second? Let's see." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 " subs(\{w=24,l=72\},[eq1,eq2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/\"#vF%/F%\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/\"#sF%/F%\"#k" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 238 "Well, yes, if ever so slightly. The left-hand side of t he second equation has decreased by 2 towards the right-hand side. We could pursue this method of guessing and then trying to improve our g uess, but let's put that off until later. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 259 34 "By Hand, as wi th pencil and paper." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 533 " You can choose to try to solve your equations by hand by which I mean to manipulate the equations the same way you would do using pencil an d paper. This can be done without using solve at all. This is what w e did in our original solution to the swimming pool problem. On the o ther hand, we could simplfy the equations and then use the word solve. For example, let's solve the two equations, eq1 and eq2, by hand. \+ Using the Maple word solve we need only solve one equation for one unk nown. Here is a possible way to proceed: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " eq3 := subs(l=solve(eq 1,l),eq2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,$%\"wG\"\"$, &F'\"\"\"\"#SF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " sol1 \+ := w = solve(eq3,w); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G/%\" wG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " sol2 := l = so lve(subs(sol1,eq1),l); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G/% \"lG\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " sol := \{sol 1, sol2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<$/%\"wG\"#?/%\"l G\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 19 "Graphical solution." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 421 "Another way to solve equation s is graphically. Here we can plot each equation, using plot or impli citplot and use the pointer to locate the approximate solutions. Thes e solution(s) are the located where the graphs of the equations coinci de. Note: the word implicitplot can only be found in Maple V Release 2. Earlier versions of Maple only plot functions. This method also \+ works very well on a graphing calculator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " with(plots): " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " implicitplot(\{eq1,eq2 \},w=10..30,l=50..70); " }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6U7$ 7$$\"#5\"\"!$\"#]F*7$$\"1++++++S5!#9$\"1++++++S]F07$7$$\"1++++++!3\"F0 $\"1++++++!3&F0F-7$F47$$\"1++++++?6F0$\"1************>^F07$7$$\"1+++++ +g6F0$\"1************f^F0F:7$F@7$$\"1*************>\"F0$\"1*********** **>&F07$7$$\"1++++++S7F0$\"1************R_F0FF7$FL7$$\"1************z7 F0$\"1++++++!G&F07$7$$\"1++++++?8F0$\"1************>`F0FR7$FX7$$\"1*** *********f8F0$\"1************f`F07$7$$\"#9F*$\"1*************R&F0Fhn7$ F^o7$$\"1************R9F0$\"1************RaF07$7$$\"1++++++![\"F0$\"1) ***********zaF0Fdo7$Fjo7$$\"1************>:F0$\"1************>bF07$7$$ \"1++++++g:F0$\"1)***********fbF0F`p7$Ffp7$$\"1*************f\"F0$\"1* ************f&F07$7$$\"1,+++++S;F0$\"1)***********RcF0F\\q7$Fbq7$$\"1+ +++++!o\"F0$\"1)***********zcF07$7$$\"1,+++++?dF0F hq7$F^r7$$\"1************f>F0$\" 1************>fF07$7$$\"1,+++++g>F0$\"1(***********ffF0F\\t7$Fbt7$$\"1 **************>F0$\"1**************fF07$7$$\"1,+++++S?F0$\"1'********* **RgF0Fht7$F^u7$$\"1************z?F0$\"1)***********zgF07$7$$\"1,+++++ ?@F0$\"1'***********>hF0Fdu7$Fju7$$\"1)***********f@F0$\"1)*********** fhF07$7$$\"1,++++++AF0$\"1'************>'F0F`v7$Ffv7$$\"1)***********R AF0$\"1)***********RiF07$7$$\"1,+++++!G#F0$\"1&***********ziF0F\\w7$Fb w7$$\"1)***********>BF0$\"1(***********>jF07$7$$\"1,+++++gBF0$\"1&**** *******fjF0Fhw7$F^x7$$\"1)************R#F0$\"1)************R'F07$7$$\" 1,+++++SCF0$\"1&***********RkF0Fdx7$Fjx7$$\"1)***********zCF0$\"1)**** *******zkF07$7$$\"1,+++++?DF0$\"1&***********>lF0F`y7$Ffy7$$\"1(****** *****fDF0$\"1)***********flF07$7$$\"1,++++++EF0$\"1%************f'F0F \\z7$Fbz7$$\"1)***********REF0$\"1)***********RmF07$7$$\"1,+++++!o#F0$ \"1%***********zmF0Fhz7$F^[l7$$\"1)***********>FF0$\"1************>nF0 7$7$$\"1-+++++gFF0$\"1%***********fnF0Fd[l7$Fj[l7$$\"1)************z#F 0$\"1(************z'F07$7$$\"1-+++++SGF0$\"1$***********RoF0F`\\l7$Ff \\l7$$\"1)***********zGF0$\"1)***********zoF07$7$$\"1-+++++?HF0$\"1$** *********>pF0F\\]l7$Fb]l7$$\"1)***********fHF0$\"1(***********fpF07$7$ $\"1-++++++IF0$\"1$*************pF0Fh]l-%'COLOURG6&%$RGBG\"\"\"F*F*-F$ 6U7$7$$\"1nmmmmmm;F0F+7$Fiq$\"1,+++++S]F07$7$$\"1LLLLLL$p\"F0F7F^_l7$7 $$\"1MLLLLL$p\"F0F77$$\"1,++++++F*$\"1************ *p&F07$7$$\"1mmmmmm1>F0FarFecl7$F[dl7$$\"1++++++?>F0Fgr7$7$$\"1LLLLLLL >F0F]sF_dl7$Fcdl7$$\"1++++++S>F0$\"1)***********>eF07$7$FctFisFgdl7$F] el7$$\"1************z>F0$\"1)***********RfF07$7$$\"1lmmmmm')>F0FetF_el 7$FeelFht7$7$$\"1KLLLLL8?F0FauFht7$Fjel7$$\"1************>?F0$\"1(**** *******fgF07$7$F_uF]vF^fl7$Fdfl7$$\"1************f?F0$\"1)***********z hF07$7$$\"1lmmmmmm?F0FivFffl7$F\\gl7$Feu$\"1(***********RiF07$7$$\"1KL LLLL$4#F0FewF`gl7$Fdgl7$$\"1*************4#F0$\"1'************H'F07$7$ F[vFaxFhgl7$F^hl7$$\"1************R@F0$\"1'***********>kF07$7$$\"1lmmm mmY@F0F]yF`hl7$Ffhl7$$\"1************f@F0$\"1(***********zkF07$7$$\"1K LLLLLt@F0FiyFjhl7$F`il7$$\"1************z@F0$\"1'***********RlF07$7$Fg vFezFdil7$Fjil7$$\"1************>AF0$\"1(***********fmF07$7$$\"1lmmmmm EAF0Fa[lF\\jl7$Fbjl7$$\"1************RAF0$\"1'***********>nF07$7$$\"1J LLLLL`AF0F]\\lFfjl7$F\\[m7$$\"1************fAF0$\"1&***********znF07$7 $FcwFi\\lF`[m7$Ff[m7$$\"1*************H#F0$\"1'*************oF07$7$$\" 1kmmmmm1BF0Fe]lFh[m7$F^\\m7$FiwF[^l7$7$$\"1JLLLLLLBF0Fa^lFb\\mFc^l-%+A XESLABELSG6$%\"wG%\"lG" 2 377 377 377 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 112 7 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {TEXT 260 14 "Using fsolve." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 445 "Another way to solve equations is with fsolve . This word e mploys methods of calculus to find floating point approximations to th e equation you are trying to solve. If you do not supply fsolve with \+ an interval in which to search for a solution, then it returns the fir st solution it finds. This word works well in conjunction with plot . You can use plot to narrow down the search interval, and then use \+ fsolve to get the 'exact' answer." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 " Problems" }}{EXCHG {PARA 289 "" 0 "" {TEXT -1 46 " A. Solve th e following systems of equations." }}{PARA 0 "" 0 "" {TEXT -1 5 "1. \+ " }{XPPEDIT 18 0 "x^2 + y^2 = 10" "/,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F' \"#5" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "y=3*x" "/%\"yG*&\"\"$\"\"\" %\"xGF&" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "2. " } {XPPEDIT 18 0 " a+3*b+c=10 , 2*a-b+72*c=20 , 7*a-3*b+21*c=30" "6 %/,(%\"aG\"\"\"*&\"\"$F&%\"bGF&F&%\"cGF&\"#5/,(*&\"\"#F&F%F&F&F)!\"\"* &\"#sF&F*F&F&\"#?/,(*&\"\"(F&F%F&F&*&\"\"$F&F)F&F0*&\"#@F&F*F&F&\"#I" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "3. " }{XPPEDIT 18 0 "x ^5 - 5*x^2 + 2 = 0 " "/,(*$%\"xG\"\"&\"\"\"*&\"\"&F'*$F%\"\"#F'!\"\"\" \"#F'\"\"!" }{TEXT -1 29 " use plot and fsolve here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "4. " }{XPPEDIT 18 0 "x-3*y+z=a , 2*x+y-3*z= b , 9*x +3*y+5*z=c" "6%/,(%\"xG\"\"\"*&\"\"$F&%\"yGF&!\"\"%\"zGF&% \"aG/,(*&\"\"#F&F%F&F&F)F&*&\"\"$F&F+F&F*%\"bG/,(*&\"\"*F&F%F&F&*&\"\" $F&F)F&F&*&\"\"&F&F+F&F&%\"cG" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "x , y" "6$%\"xG%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 85 "B. Solve the following \+ word problems by setting up and solving a system of equations." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 343 "1. The height of the Eiffel Tower in Paris is 125 feet less than twice the h eight of the Washington Monument. The latter is 75 higher than the Gr eat Pyramid of Cheops and 105 feet higher than the dome of St. Peter's Church in Rome. If the sum of the heights of these four edifices is \+ 2510 feet, find the height of each to the nearest foot. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 319 "2. A large \+ conference table is to be constructed in the shape of a rectangle with two semicircles at the ends (see figure). Find the dimensions of the table, given that the area of the rectangular portion is to be twice \+ the sum of the areas of the circular ends, and the perimeter of the wh ole table is to be 40 feet." }}{PARA 0 "" 0 "" {TEXT -1 43 "Here is a \+ diagram to accompany the problem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F:=plot([-cos(t)+1,sin(t)+1,t=Pi/2..-Pi/2]):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "G:=plot([cos(t)+6,sin(t)+1,t=Pi/2.. -Pi/2]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "H:=plot([[1,0],[6,0],[6 ,2],[1,2],[1,0]],style=LINE):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pl ots[display](\{F,G,H\},axes=none,scaling=constrained);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7S7$$\"1.^?++++5!#:\"\"!7$$\"1?#*))fsv:$* !#;$\"15h^rqoVB!#=7$$\"1$zNp+)*Gs)F/$\"1I*y)[IZ)=)F27$$\"1Of(*RVoh!)F/ $\"1r'*z&pKtOK-\"F/7$$\"13*>Po)Q0]F/$\"18kpQqjO8F/7$$\"1=6(='fGTWF/$\"1SG.DqJ(o \"F/7$$\"1d(QML[m)QF/$\"1Klru?G'3#F/7$$\"1[EoR$y;U$F/$\"11i+([a$oCF/7$ $\"1&>jET@o#HF/$\"1H*yQfV5$HF/7$$\"1X(zl8*fiCF/$\"1;#\\Ndt#GMF/7$$\"1; %oAO8#[?F/$\"1/`i@zCORF/7$$\"1VpWXLZ,=o)>RvK \"F/$\"12C-!f-7-&F/7$$\"1E@\")[spW5F/$\"1[w+'3%**\\bF/7$$\"1E9em(zEc(F =$\"1BQRJd5&='F/7$$\"1S@Se5X%Q&F=$\"1$[<*))[(Gw'F/7$$\"1RqC&>p)4MF=$\" 1%fY!*o%*3T(F/7$$\"1Q(4(fl]T>F=$\"1j3C'f^!R!)F/7$$\"1!eo5\">2X%)F2$\"1 $e)e*)e7.()F/7$$\"1Dsr#*o&pK#F2$\"13&4&y4?=$*F/7$$\"1?sFf*oA?\"!#@$\"1 \\M7KM\\%)**F/7$$\"1'[hhSv;H#F2$\"12'[6lhw1\"F*7$$\"1N(*e+X$==)F2$\"1F L-L'ew7\"F*7$$\"14(p\\@E\"f=F=$\"1J#ehQH>>\"F*7$$\"1U[b!=l8P$F=$\"1)** H5zpuD\"F*7$$\"1lo,42!RF&F=$\"12*[+7j/K\"F*7$$\"1J!epLF_](F=$\"1l]MMV4 !Q\"F*7$$\"1cb2j!e@/\"F/$\"1rv3L#*[W9F*7$$\"1@Z`+WVU8F/$\"1SuMi`Y+:F*7 $$\"13dd!*>d,zg\"F*7$$\" 1KZ!==dh[#F/$\"1b>*3)[')f;F*7$$\"14c[`48=HF/$\"14(z_+Dgq\"F*7$$\"1??7r ^*)*R$F/$\"18%4$*Qc7v\"F*7$$\"1&3Kcb(y)*QF/$\"1NuCj!3Bz\"F*7$$\"1$p%za GL[WF/$\"19(eJ4R<$=F*7$$\"1`tqe0l,]F/$\"1Oh'yn?h'=F*7$$\"1B*>IO;%*e&F/ $\"1XU.B%yu*=F*7$$\"1VQ3aTl\">'F/$\"1#=!z%=VY#>F*7$$\"1mK_&Gn(fnF/$\"1 1Oa5\"\\g%>F*7$$\"1(zU?3\\aU(F/$\"1\"=GZE!Hm>F*7$$\"1N=2S#*\\J!)F/$\"1 \")>#=gL/)>F*7$$\"1^\"f/N:go)F/$\"1t73M'H8*>F*7$$\"1-]_>z#zJ*F/$\"1Yhn w6n(*>F*7$F($\"\"#F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-F$6$7S7$$\"1(*[z* *******fF*F+7$$\"1y5,uUUogF*F07$$\"1@kI*>5x7'F*F67$$\"11C+m:$Q>'F*F;7$ $\"1V(*R`>^fiF*FA7$$\"17=/<)3PK'F*FF7$$\"1N_[.5$>Q'F*FK7$$\"1)zV^Vb1W' F*FP7$$\"14!G;8h%*\\'F*FU7$$\"1*)G\"QSreb'F*FZ7$$\"1Chlm^L6mF*Fin7$$\" 1N<.m@$yl'F*F^o7$$\"1!oL(eyJ2nF*Fco7$$\"1E?M'3SPv'F*Fho7$$\"1eJxj'y^z' F*F]p7$$\"11`XlE&)HoF*Fbp7$$\"1==8!3Ys'oF*Fgp7$$\"1(y=^FIb*oF*F\\q7$$ \"1'=MB?tV#pF*Faq7$$\"1zfT*[bh%pF*Ffq7$$\"1Hv/38!f'pF*F[r7$$\"1.HSM\\e !)pF*F`r7$$\"1K*)3G\\b\"*pF*Fer7$$\"1$G2J/tw*pF*Fjr7$$\"1T5t(z)****pF* F`s7$$\"1QQfC$3x*pF*Fes7$$\"15%*\\l\"==*pF*Fjs7$$\"1..&yt39)pF*F_t7$$ \"1^W>[jGmpF*Fdt7$$\"1K)4H*4EZpF*Fit7$$\"1>/jEx%\\#pF*F^u7$$\"1WCp$>%y &*oF*Fcu7$$\"1Gl%*flvloF*Fhu7$$\"1HC%4!G%)HoF*F]v7$$\"1iuo'G)*Rz'F*Fbv 7$$\"1F&>=G%Q^nF*Fgv7$$\"1S9l/p=3nF*F\\w7$$\"1)z()G[5+m'F*Faw7$$\"1\"z OWC@,h'F*Ffw7$$\"1J0_9n;blF*F[x7$$\"1k#HT%\\$)*\\'F*F`x7$$\"13!)pj$e5W 'F*Fex7$$\"1:;f%eM3Q'F*Fjx7$$\"1uwWrK-CjF*F_y7$$\"1@dz\"4buD'F*Fdy7$$ \"1'F*Fiy7$$\"1&3a\\Y)RJhF*F^z7$$\"1*\\Z!3s?ogF*Fcz7$Fc[lFfzFh z-F$6%7'7$$\"\"\"F+F+7$$\"\"'F+F+7$FjdlFfz7$FgdlFfzFfdlFhz-%&STYLEG6#% %LINEG-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 2 377 377 377 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20040 0 12010 0 255 0 0 255 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 468 "3. A l ocal hardware store worker is making up a fertilizer mix from some lef t over fertilizer from the summer. There are three types, with 30 per cent, 20 percent and 15 percent nitrogen content respectively. When \+ he mixed all the fertilizers together and tested the nitrogen content, he found that the mix weighed 600 pounds and contained 25 percent nit rogen content. How many pounds of each type did he have left over? H ow many solutions does this problem have?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 319 "4. A 30 foot ladder and a 40 foot ladde r are positioned so as to cross each other in an alley. That is, they are leaning up against opposite walls with their bases snug up agains t the base of the opposite walls. Given that they cross at a point 10 feet above the floor of the alley, determine the width of the alley. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "5. Make up your own algebra problem to set up and solve using Maple. It can be a variation on one of the problems above, or it can be som ething entirely different." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 " More Abo ut Plotting" }}{EXCHG {PARA 0 "" 0 "with(plots)" {TEXT -1 45 " We have already used a few words from the " }{TEXT 267 12 "with(plots);" } {TEXT -1 155 " package. When you want to learn more about any pack age (in this case with(plots);) start by looking at the commands in th e package. Do this by typing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7S%(animateG%*animate3dG%-changecoordsG%,complexplotG%. complexplot3dG%*conformalG%,contourplotG%.contourplot3dG%*coordplotG%, coordplot3dG%-cylinderplotG%,densityplotG%(displayG%*display3dG%*field plotG%,fieldplot3dG%)gradplotG%+gradplot3dG%-implicitplotG%/implicitpl ot3dG%(inequalG%-listcontplotG%/listcontplot3dG%0listdensityplotG%)lis tplotG%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%(odeplotG%'par etoG%*pointplotG%,pointplot3dG%*polarplotG%,polygonplotG%.polygonplot3 dG%.polyhedraplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-se toptions3dG%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)t extplotG%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "We have already used plot , i mplicitplot , and display (in problem 2 of part B). To learn more \+ about these very useful commands simply type" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?plot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "?implicitplot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "?display" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "?textplot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "?animate" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 263 "or anythi ng else that might interest you. Look at the bottom of the help files for examples. Sometimes, you are your best teacher. Learn by experi menting with a few (or all) of these commands. Also, you will learn m ore about these words in worksheets to come." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 " Putting in a parameter. " }}{EXCHG {PARA 0 "" 0 "parameter" {TEXT -1 238 " One of the advantages of solving a problem in a Maple worksheet is that it gives the capability of going back an d changing the numbers in the problem to study how the solution change s. In fact, you are led to the practice of putting a " }{TEXT 269 9 " parameter" }{TEXT -1 196 " into the problem. For example, to put a parameter in the swimming pool problem, we have two natural choices: \+ Replace the number 3 or the number 40 by a parameter. Let's repla ce 3 with p." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 35 "Parameterized Sw imming Pool Problem" }{TEXT -1 1 " " }}{PARA 289 "" 0 "" {TEXT 262 8 " Problem:" }{TEXT -1 175 " swimming pool is p times as long as it is w ide. It is also 40 feet longer than it is wide. What are its dimensi ons in terms of p? Describe how the dimensions vary with p." }}{PARA 0 "" 0 "" {TEXT 261 9 "Solution:" }{TEXT -1 99 " We could just go bac k to the cell containing the original equations and put in a p for 3 i n eq1. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " restart; " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "The word " }{TEXT 267 7 "restart" }{TEXT -1 44 " c lears all variables by rebooting Maple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " p := 'p';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " eq1 := l = p*w; " } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/%\"lG*&%\"pG\"\"\"%\"wGF)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " eq2 := l = w + 40;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/%\"lG,&%\"wG\"\"\"\"#SF)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Now we would solve for l and w, just as before, except now the solution is given in terms of t he parameter p. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " sol := solve(\{eq1,eq2\},\{l,w\});" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$solG<$/%\"lG,$*&%\"pG\"\"\",&!\"\"F+F*F+F-\"# S/%\"wG,$*$F,F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 245 "N ow by inspection, we can see that as p gets large, both w and l get sm all. Also, as p approaches 1 from the left, l and w get large. 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))))))))o8F\\\\l7$Fjs$FfbmF\\\\l7$F]t$\"1666666J9F\\\\l7$F`t$\"1AAAAAA i9F\\\\l7$Fct$\"1LLLLLL$\\\"F\\\\lFetFit7$-F(6$7SF+7$F.F]_n7$F2Fj`m7$F 5Fham7$F8F]\\m7$F;$\"1mmmmmmm;Fbu7$F>F\\dm7$FAFjdm7$FEFiem7$FHF^bn7$FK Febn7$FM$\"1mmmmmmmOFbu7$FPF^in7$FS$\"1LLLLLLLVFbu7$FVFju7$FYF`v7$FfnF den7$Fin$\"1nmmmmmmcFbu7$F\\o$\"1,++++++gFbu7$F_oFhw7$FboF^x7$FdoFdx7$ Fgo$F]amFbu7$Fjo$\"1mmmmmmmwFbu7$F]p$FdhmFbu7$F`p$\"1JLLLLLL$)Fbu7$Fcp $\"1kmmmmmm')Fbu7$Ffp$\"1(**************)Fbu7$FipF^`r7$F\\q$\"1jmmmmmm '*Fbu7$F_qFj[l7$FaqFa\\l7$FdqFg\\l7$Fgq$\"1*************4\"F\\\\l7$Fjq Fc]l7$F]r$\"1mmmmmmm6F\\\\l7$F`r$\"1*************>\"F\\\\l7$Fcr$\"1LLL LLLL7F\\\\l7$Ffr$\"1mmmmmmm7F\\\\l7$Fir$\"1*************H\"F\\\\l7$F\\ s$Fh_lF\\\\l7$F^s$\"1mmmmmmm8F\\\\l7$Fas$\"1*************R\"F\\\\l7$Fd s$\"1LLLLLLL9F\\\\l7$Fgs$Ff\\mF\\\\l7$Fjs$\"1+++++++:F\\\\l7$F]t$\"1LL LLLLL:F\\\\l7$F`t$\"1nmmmmmm:F\\\\l7$Fct$\"$g\"F,FetFit-%+AXESLABELSG6 $%\"wG%!G-%%VIEWG6$;F,Ffy%(DEFAULTG" 2 377 377 377 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 100 1 }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "animate" {TEXT -1 11 "The word " }{TEXT 267 7 "animate" }{TEXT -1 177 ", which is use d out of the with(plots); package, is very useful in seeing the range \+ of possible solutions for this problem. Press your pointer on play to see the animation." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 " Problems" }}{EXCHG {PARA 289 "" 0 "" {TEXT -1 178 "1. What happens to the solution to the swimming pool pr oblem as the difference of the width and length is allowed to vary? ( Keep the ratio of the width to the length constant.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 321 "2. A large conferen ce table is to be constructed in the shape of a rectangle with two sem icircles at the ends (see figure). Find the dimensions of the table, \+ given that the area of the rectangular portion is to be p times the su m of the areas of the circular ends, and the perimeter of the whole ta ble is to be 40 feet." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 153 "3. Stud y the solution to the mixture problem, no.3, in the previous problem s et. Vary a parameter of your choice and describe how the solution cha nges." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 289 "" 0 "" {TEXT -1 318 "4. A 30 foot ladder and a 40 f oot ladder are positioned so as to cross each other in an alley. That is, they are leaning up against opposite walls with their bases snug \+ up against the base of the opposite walls. Given that they cross at a point h feet above the floor of the alley, determine the width of the alley." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 " Defi ning your own Maple words." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Thi s is a good place to learn how to develop and define Maple words in a \+ worksheet. The idea is very simple:" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 264 34 " A. Name the given quantities -- " }}{PARA 0 "" 0 "" {TEXT -1 215 "As you are solving a \+ problem or developing an algorithm, assign the given quantities to app ropriate names which you have chosen. Put these assignments into a sin gle input cell, which we will call the parameter cell." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 265 36 " \+ B. Compute the desired quantity --" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 272 "Use the names assigned in your maple statements which \+ you make in your algorithm. This part will change and grow as you dev elop the definition, but try to keep all the statements in one input c ell, which we will call the definition cell. Once you get the desired output," }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 266 27 "C. Make t he Definition -- " }}{PARA 0 "" 0 "" {TEXT -1 116 "This involves mostl y choosing a name for the procedure, inserting it at the top of the de finition cell in the line\n " }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {TEXT 269 26 "name := proc(p1,p2,...,pn)" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " \n " }}{PARA 0 "" 0 "" {TEXT -1 100 "which begi ns the definition by assigning the name and declaring the input parame ters p1, p2, etc. " }}{PARA 0 "" 0 "" {TEXT -1 49 "At the bottom of \+ the definition cell is the word " }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }{TEXT 269 5 "end; " }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "which signals the end of the definition. " }}{PARA 0 "" 0 "" {TEXT -1 116 "Now, when you execute the definition cell, you should ge t a nicely formatted version of the definition as output. " }}{PARA 0 "" 0 "" {TEXT -1 145 "This is not the only set of steps you could fo llow when developing a definition, but it is very natural one and work s well for small definitions." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "to define a word" {TEXT -1 37 "By way of example, suppos e we wanted " }{TEXT 269 16 "to define a word" }{TEXT -1 81 " swim which would return the solution to the swimming pool problem above. \+ " }}{PARA 0 "" 0 "" {TEXT -1 76 "Copy all of the input cells used to obtain the solution into one input cell." }}{PARA 0 "" 0 "" {TEXT -1 106 " Then all we need to do to define the word is to insert a proc li ne at the top and an end at the bottom. " }}{PARA 0 "" 0 "" {TEXT -1 55 "When inserting the proc line, you must decide what the " }{TEXT 269 6 "inputs" }{TEXT -1 215 " for the word will be. We will let the ratio of the length of the pool to the width of the pool be the \+ only input in this word. We were calling that p in the equations, so \+ use the same name in the proc line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " swim := proc(p) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " eq1 := l = p*w; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " eq2 := l = w + 40; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " sol := solve(\{eq1,eq2\},\{l,w\}); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " end;" }}{PARA 7 "" 1 "" {TEXT -1 43 "Wa rning, `eq1` is implicitly declared local" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, `eq2` is implicitly declared local" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, `sol` is implicitly declared local" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%%swimG:6#%\"pG6%%$eq1G%$eq2G%$solG6\"F,C%>8$/% \"lG*&9$\"\"\"%\"wGF4>8%/F1,&F5F4\"#SF4>8&-%&solveG6$<$F/F7<$F1F5F,F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 272 "When you execute the input cell containing the defi nition, the word swim has been added to the vocabulary and can be u sed like any other Maple word. The assignments made in the definitio n are declared local and a warning is issued unless you declare the na mes either " }{TEXT 269 15 "local or global" }{TEXT -1 140 ". Usua lly, you will want any assignments to be local, just in case you are \+ using the same name outside the word to mean something else." }}{PARA 0 "" 0 "" {TEXT -1 168 "The output from a word is the output from the \+ last line before the end line. So, for example, if the pool is 3 tim es as long as it is wide then p = 3 and we would say" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " swim(3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"wG\"#?/%\"lG\"#g" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "On the other hand, if we put in a ridiculous input what \+ would happen?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " swim(- 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"wG!\")/%\"lG\"#K" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "You can redefine the word if there is something that needs change d. For example, the swimming pool problem doesn't really have a solut ion if p is not positive, so we could insert an " }{TEXT 269 10 "error trap" }{TEXT -1 9 " here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " swim := proc(p) " } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " if not type(p,name) and not p \+ > 0 then ERROR(`oops`) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ else eq1 := l = p*w; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " eq2 := l = w + 40; " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " sol := solve(\{eq1 ,eq2\},\{l,w\}); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " fi \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " end;" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, `eq1` is implicitly declared local" }}{PARA 7 " " 1 "" {TEXT -1 43 "Warning, `eq2` is implicitly declared local" }} {PARA 7 "" 1 "" {TEXT -1 43 "Warning, `sol` is implicitly declared loc al" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%swimG:6#%\"pG6%%$eq1G%$eq2G%$ solG6\"F,@%45-%%typeG6$9$%%nameG2\"\"!F3-%&ERRORG6#%%oopsGC%>8$/%\"lG* &F3\"\"\"%\"wGFA>8%/F?,&FBFA\"#SFA>8&-%&solveG6$<$F=FD<$F?FBF,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Now redefine swim and che ck --" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " swim(-4);" }}{PARA 8 "" 1 "" {TEXT -1 21 "Error, \+ (in swim) oops" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 460 "Usually, one doesn't spend a gre at deal of time inserting error traps in word definitionst. There are much more interesting things to do with the mathematics of the situat ion. For example, suppose we wanted to change the inputs to allow fo r changing the 40, the amount the length exceeds the width, that occu rs in the problem. Then simply copy down the definition into a new \+ input cell and make the appropriate changes. Something like this wil l work:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "swim := proc (p,excess) \nl ocal eq1, eq2, sol;\n eq1 := l = p*w; \neq2 := l = w+excess; \nsol := \+ solve(\{eq1, eq2\},\{l, w\})\n end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%%swimG:6$%\"pG%'excessG6%%$eq1G%$eq2G%$solG6\"F-C%>8$/%\"lG*&9$\"\" \"%\"wGF5>8%/F2,&F6F59%F5>8&-%&solveG6$<$F0F8<$F2F6F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "swim(3,40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"lG\"#g/%\"wG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "swim(3,ex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\" wG,$%#exG#\"\"\"\"\"#/%\"lG,$F'#\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "Notice that you can 'read off' the solution in words if you put in a variable for the excess. So, if the pool is 3 times as long as it is wide, then the width must be 1/2 of the excess and the \+ length is 3/2 of the excess." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "visual checking of answers" {TEXT 269 31 "Visual Checking of a nswers. " }}{PARA 0 "" 0 "" {TEXT -1 291 "You can also define a wor d to draw a picture of the pool. These types of words are very usef ul to perform a visual check of computations. Sometimes, you have sol ved a problem incorrectly, but do not discover that until you have per formed a visual check on the solution you have obtained." }}{PARA 0 " " 0 "" {TEXT -1 116 "To develop a visual check of our swimming pool pr oblem, we can first draw one picture. Get the dimensions of a pool" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dims := swim(3,40);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dimsG<$/%\"lG\"#g/%\"wG\"#?" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Now we want to draw a rectangle w hich is 20 by 60. We can set up a general rectangle with one corner \+ at the origin and the opposite corner at [l,w] " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rect := [[0,0],[l,0],[l,w],[0,w]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rectG7&7$\"\"!F'7$%\"lGF'7$F)%\"wG7$F'F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pool := subs(dims,rect) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%poolG7&7$\"\"!F'7$\"#gF'7$F)\" #?7$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plots[polygonpl ot](pool,\ncolor=blue,style=patch, scaling = constrained);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%)POLYGONSG6#7&7$\"\"!F(7$$\"#gF(F(7$F*$\"#?F( 7$F(F--%&STYLEG6#%&PATCHG-%(SCALINGG6#%,CONSTRAINEDG-%'COLOURG6&%$RGBG F(F($\"*++++\"!\")" 2 330 330 330 2 0 1 0 2 6 0 4 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "Now we can define the word. Copy down and join the appropriate i nput cells. Then insert a proc line at the top (deciding on what the \+ inputs will be), and an end line at the bottom of the new input cell. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "drawpool := proc(p,exce ss)\ndims := swim(p,excess);\nrect := [[0,0],[l,0],[l,w],[0,w]];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pool := subs(dims,rect);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plots[polygonplot](pool,\ncolor=blue,styl e=patch, scaling = constrained);\nend;" }}{PARA 7 "" 1 "" {TEXT -1 44 "Warning, `dims` is implicitly declared local" }}{PARA 7 "" 1 "" {TEXT -1 44 "Warning, `rect` is implicitly declared local" }}{PARA 7 " " 1 "" {TEXT -1 44 "Warning, `pool` is implicitly declared local" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)drawpoolG:6$%\"pG%'excessG6%%%dimsG %%rectG%%poolG6\"F-C&>8$-%%swimG6$9$9%>8%7&7$\"\"!F:7$%\"lGF:7$F<%\"wG 7$F:F>>8&-%%subsG6$F0F7-&%&plotsG6#%,polygonplotG6&FA/%&colorG%%blueG/ %&styleG%&patchG/%(scalingG%,constrainedGF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "drawpool(3,40);" }}{PARA 13 "" 1 "" {INLPLOT "6& -%)POLYGONSG6#7&7$\"\"!F(7$$\"#gF(F(7$F*$\"#?F(7$F(F--%&STYLEG6#%&PATC HG-%(SCALINGG6#%,CONSTRAINEDG-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")" 2 330 330 330 2 0 1 0 2 6 0 4 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 " " 0 "insequnce=true" {TEXT -1 233 "Now we can make an animation of the how the pool dimensions change as we vary the excess and/or the ratio of length to width. Make a list of the outputs from drawpool (call i t movie, say) and then use plots[display] with the option " }{TEXT 267 17 "insequence = true" }{TEXT -1 6 ". " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "movie := [seq(drawpool(3,10*i),i=1..8)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plots[display](movie,inseque nce=true);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%(ANIMATEG6*7$-%)POLYGONSG 6%7&7$\"\"!F,7$$\"#:F,F,7$F.$\"\"&F,7$F,F1-%'COLOURG6&%$RGBGF,F,$\"*++ ++\"!\")-%&STYLEG6#%&PATCHG-%(SCALINGG6#%,CONSTRAINEDG7$-F(6%7&F+7$$\" #IF,F,7$FH$\"#5F,7$F,FKF4F;F?7$-F(6%7&F+7$$\"#XF,F,7$FSF.7$F,F.F4F;F?7 $-F(6%7&F+7$$\"#gF,F,7$Ffn$\"#?F,7$F,FinF4F;F?7$-F(6%7&F+7$$\"#vF,F,7$ Fao$\"#DF,7$F,FdoF4F;F?7$-F(6%7&F+7$$\"#!*F,F,7$F\\pFH7$F,FHF4F;F?7$-F (6%7&F+7$$\"$0\"F,F,7$Fep$\"#NF,7$F,FhpF4F;F?7$-F(6%7&F+7$$\"$?\"F,F,7 $F`q$\"#SF,7$F,FcqF4F;F?" 2 330 330 330 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 46 111 1 0 0 0 100 0 }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "movie" {TEXT -1 5 "This " }{TEXT 269 5 "movi e" }{TEXT -1 255 " merely shows the dimensions of the pool increas ing linearly with the excess, as we already knew, so not much is gain ed from observing the animation. Often, however, one can learn a lot \+ from animations, if some care is taken with their construction." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 12 " Problems " }}{EXCHG {PARA 289 "" 0 "" {TEXT -1 106 " Exercise: Make a movie of the change in the pool as the ratio p chang es from 1 to 4 by increments of 1/2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 289 "" 0 "" {TEXT -1 114 "Exercise: What is wrong with t aking the original word swim and changing to proc line to read swim : = proc(p,w) ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 289 "" 0 " " {TEXT -1 129 "Exercise: Modify drawpool so that if the ratio p is \+ greater than 2, the color of the pool is blue, otherwise the color is \+ red. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 116 " Exercise: Define a word quadform to take three numbers a, b, c and return the roots of the quadratic equation " }{XPPEDIT 18 0 " \+ a*x^2 + b*x + c = 0 " "/,(*&%\"aG\"\"\"*$%\"xG\"\"#F&F&*&%\"bGF&F(F&F& %\"cGF&\"\"!" }{TEXT -1 102 ", if they are real, and the message 'no r eal roots' if they are complex. Don't just use solve here ." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 117 "Exercise : Modify the word swim so that it simply returns the point [l,w] rat her than the solution to the equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 289 "" 0 "" {TEXT -1 135 "Exercise: Define a word which takes a function and tabulates its value over a given interval with a given increment between x values." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Table of Contents" 1 "hand0.mws" "" }{HYPERLNK 17 "" 1 "mprobs.mws" "" }}}}{MARK "0 7 12 0 2" 152 } {VIEWOPTS 1 1 0 1 1 1803 }