{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 " Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 0 "" 0 "" {TEXT 263 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 262 22 " Solution Word Problems" }}{PARA 0 "" 0 "" {TEXT -1 577 "\nSolution wor d problems involve mixing of two quantities to obtain resulting quanti ty, where each of these quantitites is a solution or alloy containing \+ varying strengths of the undelying material. Some examples are salt so lutions, alcholol solutions, acid solutons, metal alloys, etc. These p roblems are similar to mixture problems in that two quantities are bei ng mixed. The main difference is that mixture problems involve externa l rates such as price per pound, while solution problems always involv e internal rates, such as the percentage of salt in a salt water solut ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 153 "[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in th is worksheet.]" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1056 "MixturePlot := proc( a, \+ ar, b, br)\n local c,ct,cr,g,as,bs,cs, spacer, bo, co, pa, pb, pc1, pc2, tp, te, r, ch;\n\n c := a + b; ct := a*ar + b*br; \+ cr := ct/c;\n g := 1.6;\n as := sqrt( a/g); bs := sqrt( b/g ); cs := sqrt( c/g);\n spacer := (as+bs+cs)/10; \n \n pa := polygonplot( [ [0,0],[as,0],[as,as*g],[0,as*g] ], \n \+ color = gold, axes = none):\n bo := as + spacer; \n \+ \n pb := polygonplot( [ [bo, 0],[bo+bs, 0],[bo+bs,bs*g],[bo,bs*g] \+ ], \n color = green, axes = none):\n co := \+ bo + spacer*1.5 + bs; \n r := a/(a+b); ch := r*cs*g;\n pc1 \+ := polygonplot( [ [co,0],[co+cs,0],[co+cs,ch],[co,ch] ], \n \+ color = gold, axes = none):\n pc2 := polygonplot( [ [ co,ch],[co+cs,ch],[co+cs,cs*g],[co,cs*g] ], \n \+ color = green, axes = none):\n\n tp := textplot( [ as + spacer/2, \+ max(as,bs)*g/2, `+`]);\n te := textplot( [ bo + bs + spacer/2, max (as,bs)*g/2, `=`]);\n\n display( pa, pb, pc1, pc2, tp, te)\n end p roc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1374 "SolutionPlot := proc( a, ar, b, br) #||a amount|a rate|b amount|b rate||\n lo cal c,ct,cr,g,as,bs,cs,spacer,bo,co,pa1,pa2,pb1,pb2,pc1,pc2,tp,te, r,ch,h;\n\n c := a + b; ct := a*ar + b*br; cr := ct/c;\n \+ g := 1.6;\n as := sqrt( a/g); bs := sqrt( b/g); cs := sqrt( c/ g);\n spacer := (as+bs+cs)/10; \n \n h := ar*as*g;\n \+ pa1 := polygonplot( [ [0,0],[as,0],[as,h],[0,h] ], \n \+ color = orange, axes = none):\n pa2 := polygonplot( [ [0, h],[as,h],[as,as*g],[0,as*g] ], \n color = tan, axes = none):\n\n bo := as + spacer; \n h := br*bs*g;\n \+ pb1 := polygonplot( [ [bo, 0],[bo+bs, 0],[bo+bs,h],[bo,h] ], \n \+ color = orange, axes = none):\n pb2 := poly gonplot( [ [bo, h],[bo+bs, h],[bo+bs,bs*g],[bo,bs*g] ], \n \+ color = tan, axes = none):\n co := bo + spacer*1.5 \+ + bs; \n \n h := cr*cs*g; \n pc1 := polygonplot( [ [co,0],[ co+cs,0],[co+cs,h],[co,h] ], \n color = oran ge, axes = none):\n pc2 := polygonplot( [ [co,h],[co+cs,h],[co+cs, cs*g],[co,cs*g] ], \n color = tan, axes = no ne):\n\n tp := textplot( [ as + spacer/2, max(as,bs)*g/2, `+`]);\n te := textplot( [ bo + bs + spacer/2, max(as,bs)*g/2, `=`]);\n\n \+ display(pa1,pa2, pb1, pb2, pc1, pc2, tp, te)\n end proc:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 537 "MixtureTableTest := proc( a, ar, b, br )\n # a = quanity of a, ar = a's rate, and so forth \+ for b\n local rt, M ;\n rt := (a*ar + b*br)/(a+b);\n if ( not( (whattype(rt) = integer) or\n (whattype(rt) = f loat) or\n (whattype(rt) = fraction) )) then rt := `? ` fi;\n array( [ [``, `Quantity *`,` Rate`,` = Result`], \n \+ [ `A`, a, ar, a*ar], \n [ `B `, b, br, b*br], \n [ `A+B `, a+b, rt , a*ar + b*br] ]);\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 692 "MixtureTable1 := proc( a, ar, b, br, cr )\n # \+ a = quanity of a, ar = a's rate, and so forth for b\n local rt, M, equation;\n rt := (a*ar + b*br)/(a+b);\n if( not( (whatty pe(rt) = integer) or\n (whattype(rt) = float) or\n \+ (whattype(rt) = fraction) )) then rt := `?` fi;\n M : = array( [ [``, `Quantity *`,` Rate`,` = Result`], \n \+ [ `A`, a, ar, a*ar], \n [ `B`, b, \+ br, b*br], \n [ `A+B `, a+b, cr, (a+b)*cr ] ]);\n print(M);\n print(` `):\n equation := M[2,4] + M[ 3,4] = M[4,4];\n print( equation );\n solve( equation, x); \+ print(x = %);\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 624 "MixtureTableEq := proc( ar, br, cr, c) #||a rate|b rate|c rate|c amount||\n \n \+ local a, b, rt, M, equation;\n rt := cr ; #(a*ar + b*br)/(a +b);\n a := x; b := c-a;\n \n M := array( [ [``, `Q uantity *`, ` Rate`,` = Result`], \n [ `A`, a, \+ ar, a*ar ], \n [ `B`, b, br, (c- a)*br ], \n [ `A+B `, c, cr, c*cr ] ] );\n print(M);\n print(` `):\n equation := M[2,4] + M[3,4 ] = M[4,4];\n print( equation );\n solve( equation, x); print (` `); print(A = %, B = c-%); print(` `);\n \n end proc:\n\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "MixtureTableSoln := proc( ar, br, cr, c) #||a rate|b rate|c rate|c amount|| \n local \+ a, rt, M, equation;\n rt := cr ; #(a*ar + b*br)/(a+b);\n a : = solve( x*ar + (c-x)*br = c*cr, x);\n \n array( [ [``, ` Quantity *`, ` Rate`,` = Result`], \n [ `A`, a, ar, a*ar ], \n [ `B`, c-a, br, \+ (c-a)*br ], \n [ `A+B `, c, cr, c*cr \+ ] ]);\n \n end proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "1. Gene ral Idea" }}{PARA 0 "" 0 "" {TEXT -1 312 "\nSolution problems always i nvolve some component of larger amounts. When you add two quantities w ith different strengths, what is the resulting strength of the result? Lets get a rough idea of what is going on.\n\n_______________________ ______________________________________________________________________ _____\n\n" }{TEXT 258 13 "Problem 1.1 :" }{TEXT -1 245 " You combine 1 2 gallons of a 15% acid solution with 34 gallons of a 22% acid solutio n. \n 1. How many gallons are in the mixture?\n 2. How m uch total acid is in the mixture?\n 3. What is the strength of \+ the acid in the mixture?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "`Total number of gallons` := 12 + 34;\n`Total amount of acid` := 12*. 15 + 34*.22;\n`Strenth of Acid in mixture` = evalf( %/ %%, 5);" }}} {PARA 0 "" 0 "" {TEXT -1 305 "\nHere is a diagram which shows what is \+ happening. The red part at bottom is the acid. Note that the amount of acid in the two containers at left equals the amount of acid in the r ightmost container. (It may appear no higher because the rightmost con tainer is wider than either of the left two containers.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " SolutionPlot( 12, .15, 34, .22 ); " }}}{PARA 0 "" 0 "" {TEXT -1 102 "\n\n\n_____________________________ ____________________________________________________________________\n \n" }{TEXT 265 13 "Problem 1.2 :" }{TEXT -1 88 " You combine 100 gallo ns of a 10% acid solution with 5 gallons of a 80% acid solution. \n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "SolutionPlot( 100, .10, 5, \+ .80 );" }}}{PARA 0 "" 0 "" {TEXT -1 102 "\n\n_________________________ ______________________________________________________________________ ___\n\n" }{TEXT 261 13 "Problem 1.3 :" }{TEXT -1 177 " 20 kg of a 13% \+ titanium alloy and 34 kg of a 60% titanium alloy are melted and mixed together to form a new material. What is concentration of titanium in the resulting alloy?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Sol utionPlot( 20, .13, 34, .60 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 64 "2. Mixture Problems - Solutions \+ (only one known quantity)" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 256 13 "Problem 2.1 :" }{TEXT -1 124 " How many quarts of a solu tion which is 3% salt should be added to 30 quarts of a 9% salt soluti on to obtain a 7% solution?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "MixtureTable1( x, .03, 30, .09, .07); #||a amount|a rate|b am ount|b rate|c rate||" }}}{PARA 0 "" 0 "" {TEXT -1 41 "\nLets see a pic ture of what is happening." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "SolutionPlot( 15, .03, 30, .09 ); #||a amount||a rate|b amount|b \+ rate||" }}}{PARA 0 "" 0 "" {TEXT -1 101 "\n___________________________ ______________________________________________________________________ _\n\n" }{TEXT 260 13 "Problem 2.2 :" }{TEXT -1 214 " How many liters o f pure orange juice should be added to 12 liters of a 10% alcholic bev erage to create a 3% alcoholic beverage?\n\n\n(Hint : pure orange juic e has no alchohol, so its concentration of alcohol is 0%);\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "MixtureTable1( x, 0, 12, .10 , .02); #||a amount|a rate|b amount|b rate|c rate|| rate;\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "SolutionPlot( 48, 0, 12, .10 ); #||a amount||a rate|b amount|b rate||" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 64 "3. Mixture Problems - \+ Solutions (both quantities unknown)" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 257 13 "Problem 3.1 :" }{TEXT -1 109 " How many liter s of 40% alcohol solution A with 8% alcohol solution B to create 30 li ters of a \n25% solution?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "MixtureTableEq( .42, .08, .25, 30); #||a rate|b rate|c rate|c am ount|| " }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nHere is a view of the compl ete situation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "MixtureTab leSoln( .42, .08, .25, 30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "SolutionPlot( 15, .42, 15, .08); #||a amount||a rate|b amount|b r ate||" }}}{PARA 0 "" 0 "" {TEXT -1 101 "\n____________________________ ______________________________________________________________________ \n\n" }{TEXT 259 13 "Problem 3.2 :" }{TEXT -1 125 " How many kilograms of a 90% aluminum alloy should be melted with a with 70% aluminum all oy to create 200 kg of an 82% alloy?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "MixtureTableEq( .90, .70, .82, 200); #||a rate|b rate|c rate|c amount|| \n" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nHere is \+ a view of the complete situation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "MixtureTableSoln( .90, .70, .82, 200); #||a rate |b rate|c rate|c amount|| " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "SolutionPlot( 120, .70, 80, .82); #||a amount||a rate|b amount|b \+ rate||" }}}{EXCHG }{EXCHG }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 36 "\n \251 2 002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 " 18 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }