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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 221 169 169 {PLOTDATA 2 "6,-%'CU RVESG6$7U7$$\"\"\"\"\"!$F*F*7$$\"+8q9@**!#5$\"+OBL`7F/7$$\"+6;$eo*F/$ \"+t))*o[#F/7$$\"+f[w(H*F/$\"+FbC\"o$F/7$$\"++o1j()F/$\"+Tn`<[F/7$$\"+ V*p,4)F/$\"+CD&y(eF/7$$\"+tio*G(F/$\"+g5ZXoF/7$$\"+'*)RUP'F/$\"+HC80xF /7$$\"+^zEe`F/$\"+a#zKW)F/7$$\"+=HzdUF/$\"+B0F[!*F/7$$\"+Q*p,4$F/$\"+l ^c5&*F/7$$\"+UJ\"Q(=F/$\"+3D(G#)*F/7$$\"+D>0zi!#6$\"+%Gn-)**F/7$$!+m>0 ziFaoFbo7$$!+YJ\"Q(=F/$\"+2D(G#)*F/7$$!+U*p,4$F/$\"+j^c5&*F/7$$!+8HzdU F/$\"+E0F[!*F/7$$!+YzEe`F/$\"+d#zKW)F/7$$!+$*)RUP'F/$\"+JC80xF/7$$!+$G 'o*G(F/$\"+]5ZXoF/7$$!+]*p,4)F/$\"+9D&y(eF/7$$!+0o1j()F/$\"+Kn`<[F/7$$ !+i[w(H*F/$\"+>bC\"o$F/7$$!+8;$eo*F/$\"+l))*o[#F/7$$!+9q9@**F/$\"+IBL` 7F/7$$!\"\"F*$!+:w1-T!#>7$$!+8q9@**F/$!+QBL`7F/7$$!+6;$eo*F/$!+t))*o[# F/7$$!+f[w(H*F/$!+FbC\"o$F/7$$!+,o1j()F/$!+Sn`<[F/7$$!+Y*p,4)F/$!+?D&y (eF/7$$!+xio*G(F/$!+c5ZXoF/7$$!+())RUP'F/$!+PC80xF/7$$!+RzEe`F/$!+i#zK W)F/7$$!+1HzdUF/$!+H0F[!*F/7$$!+M*p,4$F/$!+m^c5&*F/7$$!+QJ\"Q(=F/$!+4D (G#)*F/7$$!+%)=0ziFao$!+&Gn-)**F/7$$\"+2?0ziFao$!+%Gn-)**F/7$$\"+]J\"Q (=F/$!+2D(G#)*F/7$$\"+Y*p,4$F/$!+i^c5&*F/7$$\"+ " 0 "" {MPLTEXT 1 0 16 "evalf(2/(5*Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W&RKF\"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 23 "In a similar way, when " }{XPPEDIT 18 0 "r=100" "6#/%\" rG\"$+\"" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "dr/dt = 1/(2*Pi*r);" "6#/*&%#drG\"\"\"%#dtG!\"\"*&F &F&*(\"\"#F&%#PiGF&%\"rGF&F(" }{TEXT -1 1 " " }{TEXT 301 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "dA/dt" "6#*&%#dAG\"\"\"%#dtG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "8/(2*Pi*`.`*100)=1/(25*Pi)" "6#/*&\"\")\"\"\"* *\"\"#F&%#PiGF&%\".GF&\"$+\"F&!\"\"*&F&F&*&\"#DF&F)F&F," }{TEXT -1 1 " " }{TEXT 299 1 "~" }{TEXT -1 16 " 0.01273239544. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Thus the rate of change o f the radius when the radius is 100 metres is " }{XPPEDIT 18 0 "1/(25 *Pi);" "6#*&\"\"\"F$*&\"#DF$%#PiGF$!\"\"" }{TEXT -1 1 " " }{TEXT 300 1 "~" }{TEXT -1 31 " 0.01273239544 metres per sec." }}{PARA 0 "" 0 " " {TEXT -1 34 "This gives the answer to part (b)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(1/(25* Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W&RKF\"!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 90 ": We can investigat e the results obtained here in terms of small changes in the variables ." }}{PARA 0 "" 0 "" {TEXT -1 89 "We estimate the change in the radius during 0.1 secs. after the radius reaches 10 metres." }}{PARA 0 "" 0 "" {TEXT -1 14 "During this 0." }{XPPEDIT 18 0 "1 =1/10" "6#/\"\"\"*&F $F$\"#5!\"\"" }{TEXT -1 31 " secs. the area increases by 0." } {XPPEDIT 18 0 "8 =4/5" "6#/\"\")*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 89 " square metres, since the area increases at the constant rate of 8 sq uare metres per sec." }}{PARA 0 "" 0 "" {TEXT -1 57 "This means that t he area of the oil slick increases from " }{XPPEDIT 18 0 "100*Pi" "6#* &\"$+\"\"\"\"%#PiGF%" }{TEXT -1 18 " square metres to " }{XPPEDIT 18 0 "100*Pi;" "6#*&\"$+\"\"\"\"%#PiGF%" }{TEXT -1 22 " + 0.8 square metr es. " }}{PARA 0 "" 0 "" {TEXT -1 23 "To find the new radius " }{TEXT 303 1 "r" }{TEXT -1 24 " we solve the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*r^2 = 100*Pi;" "6#/*&%#PiG\"\"\"*$% \"rG\"\"#F&*&\"$+\"F&F%F&" }{TEXT -1 8 " + 0.8. " }}{PARA 0 "" 0 "" {TEXT -1 39 "Dividing both sides of the equation by " }{XPPEDIT 18 0 " Pi" "6#%#PiG" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "r^2=100+4/(5*Pi)" "6#/*$%\"rG\"\"#,&\"$+\"\"\"\"*&\" \"%F)*&\"\"&F)%#PiGF)!\"\"F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=``" "6#/%\"rG%!G" }{TEXT -1 1 " " }{TEXT 305 1 "+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(100+4/(5*Pi))" "6#-%%sqrtG6#,&\"$+\"\"\"\"*&\" \"%F(*&\"\"&F(%#PiGF(!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "It is appropriate to take the positive value: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(100+4/(5*Pi))" "6#/% \"rG-%%sqrtG6#,&\"$+\"\"\"\"*&\"\"%F**&\"\"&F*%#PiGF*!\"\"F*" }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 14 " 10.01272430. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Thus we see that the change in the radius is approximately 0.01272430 metres. " }}{PARA 0 "" 0 "" {TEXT -1 46 "Since this change occurs over an interval of " } {XPPEDIT 18 0 "1/10;" "6#*&\"\"\"F$\"#5!\"\"" }{TEXT -1 173 " sec, the rate of change of the radius with respect to time is approximately 0. 1272430 metres per sec. This agrees to 3 figures with the answer obtai ned above for part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "r1 := 10;\n[solve(Pi*r^2=100*Pi+0.8 ,r)];\nr2 := op(remove(_u->_u<0,%));\n(r2-r1)/(.1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#r1G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+I CF,5!\")$!+ICF,5F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G$\"+ICF,5! \")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(ICF\"!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "Taking the positive value " }{TEXT 302 1 "r" }{TEXT -1 15 " = 10.01272430 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 94 "In a similar way, during 0.1 secs after t he radius reaches 100 metres the area increases from " }{XPPEDIT 18 0 "10000*Pi" "6#*&\"&++\"\"\"\"%#PiGF%" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "10000*Pi" "6#*&\"&++\"\"\"\"%#PiGF%" }{TEXT -1 7 " + 0.8." }}{PARA 0 "" 0 "" {TEXT -1 15 "The new radius " }{TEXT 304 1 "r" }{TEXT -1 41 " can be obtained by solving the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*r^2 = 10000*Pi;" "6#/*&%#PiG\"\"\"*$%\"rG \"\"#F&*&\"&++\"F&F%F&" }{TEXT -1 8 " + 0.8. " }}{PARA 0 "" 0 "" {TEXT -1 13 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r =`` " "6#/%\"rG%!G" }{TEXT 307 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(10000+4/(5*Pi))" "6#-%%sqrtG6#,&\"&++\"\"\"\"*&\" \"%F(*&\"\"&F(%#PiGF(!\"\"F(" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 32 "Again taking the positive value " }{XPPEDIT 18 0 "r=sqrt( 10000+4/(5*Pi))" "6#/%\"rG-%%sqrtG6#,&\"&++\"\"\"\"*&\"\"%F**&\"\"&F*% #PiGF*!\"\"F*" }{TEXT -1 1 " " }{TEXT 308 1 "~" }{TEXT -1 2 " " } {XPPEDIT 18 0 "100.0012732;" "6#-%&FloatG6$\"+KF,+5!\"(" }{TEXT -1 75 ", we see that the change in the radius is approximately 0.0012732 me tres. " }}{PARA 0 "" 0 "" {TEXT -1 171 "Hence the rate of change of th e radius with respect to time is approximately 0.012732 metres per sec . This agrees to 5 figures with the answer obtained above for part (b) ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "r1 := 100;\n[solve(Pi*r^2=10000*Pi+0.8,r)];\nr2 := op (remove(_u->_u<0,%));\n(r2-r1)/(.1);;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+KF,+5!\"($!+K F,+5F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G$\"+KF,+5!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&KF\"!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Example 2 .. sawdust pile " }}{PARA 0 " " 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 45 "Sawdust is falling onto a pile at a rate of " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 204 " cubic metre per mi n. If the sawdust maintains the shape of right circular cone with heig ht equal to the radius of the base, how fast is the height of the pile increasing when the height reaches 3 metres." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 64 "You can play the follow ing animation to illustrate this problem." }}{PARA 0 "" 0 "" {TEXT -1 91 "(Click on the graphic and use the controls in the context bar, or \+ use the animation menu.) 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7%F^ghoFaghoFbjho7%FaghoFdghoFbjho7%FdghoFgghoFbjho7%FgghoFjghoFbjho7% FjghoF]hhoFbjho7%F]hhoF_hhoFbjho7%F_hhoFbhhoFbjho7%FbhhoFehhoFbjho7%Fe hhoFhhhoFbjho7%FhhhoF[ihoFbjho7%F[ihoF\\ihoFbjho7%F\\ihoF_ihoFbjho7%F_ ihoFbihoFbjho7%FbihoFeihoFbjho7%FeihoFfihoFbjho7%FfihoFgihoFbjho7%Fgih oFhihoFbjho7%FhihoFiihoFbjho7%FiihoFjihoFbjho7%FjihoF[jhoFbjho7%F[jhoF \\jhoFbjho7%F\\jhoF]jhoFbjho7%F]jhoF^jhoFbjho7%F^jhoF_jhoFbjho7%F_jhoF `jhoFbjho7%F`jhoFhchoFbjho-%+LIGHTMODELG6#%(LIGHT_2G-%&STYLEG6#%,PATCH NOGRIDGFdco-%+PROJECTIONG6%$!$q\"F2$\"#nF2F1" 1 3 0 1 10 0 2 6 4 1 2 1.000000 67.000000 -171.000000 1 0 "Curve 1" "Curve 2" }}}{PARA 0 "" 0 "" {TEXT -1 89 "Note that rate of increase of the height of the pile decreases as the pile grows in size." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 264 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 43 "Let the radius of the conical pile at time " } {TEXT 276 1 "t" }{TEXT -1 10 " mins. be " }{TEXT 275 1 "r" }{TEXT -1 31 " metres, and let its height be " }{TEXT 277 1 "h" }{TEXT -1 9 " me tres. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Note that equation (i) relates the variables " }{TEXT 270 1 "V " }{TEXT -1 5 " and " }{TEXT 272 1 "h" }{TEXT -1 62 ", whereas equatio n (ii) relates the corresponding derivatives " }{XPPEDIT 18 0 "dV/dt" "6#*&%#dVG\"\"\"%#dtG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dh/dt; " "6#*&%#dhG\"\"\"%#dtG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Thus equation (ii) " }{TEXT 259 27 "relates the rates of \+ change" }{TEXT -1 23 " (with respect to time " }{TEXT 271 1 "t" } {TEXT -1 23 ") of the two variables " }{TEXT 269 1 "V" }{TEXT -1 5 " a nd " }{TEXT 273 1 "h" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 117 "Equation (ii) can also be obtained simply by differentiating both sides of equation (i) with respect to the variable " }{TEXT 330 1 "t " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can use equation (ii) to solve the problem, since we a re to consider the situation when " }{XPPEDIT 18 0 "h=3" "6#/%\"hG\" \"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "dV/dt" "6#*&%#dVG\"\"\"%#dtG!\"\"" }{TEXT -1 24 " has \+ the constant value " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 11 ", so, when " }{XPPEDIT 18 0 "h=3" "6#/%\"hG\"\"$" }{TEXT -1 24 ", equation (ii) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/2 = Pi*`.`*3^2;" "6#/*&\"\"\"F%\"\"#!\"\"*(%#PiGF%%\" .GF%\"\"$F&" }{TEXT 278 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "dh/dt" " 6#*&%#dhG\"\"\"%#dtG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dh /dt = 1/(2*`.`*Pi*`.`*3^2);" "6#/*&%#dhG\"\"\"%#dtG!\"\"*&F&F&*,\"\"#F &%\".GF&%#PiGF&F,F&\"\"$F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/( 18*Pi);" "6#/%!G*&\"\"\"F&*&\"#=F&%#PiGF&!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "It is us ual practice to omit mentioning the units when working through the det ails of the problem, but the final answer should be given using the ap propriate units." }}{PARA 0 "" 0 "" {TEXT -1 14 "Here the time " } {TEXT 279 1 "t" }{TEXT -1 60 " is measured in minutes and lengths are \+ measured in metres. " }}{PARA 0 "" 0 "" {TEXT -1 30 "This means that o ur value of " }{XPPEDIT 18 0 "dh/dt" "6#*&%#dhG\"\"\"%#dtG!\"\"" } {TEXT -1 32 " is measured in metres per min." }}{PARA 0 "" 0 "" {TEXT -1 70 "Thus the rate of change of the height when the height is \+ 3 metres is " }{XPPEDIT 18 0 "1/(18*Pi);" "6#*&\"\"\"F$*&\"#=F$%#PiGF $!\"\"" }{TEXT -1 1 " " }{TEXT 274 1 "~" }{TEXT -1 31 " 0.01768388256 metres per min." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(1/(18* Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+c#)Qo " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Example 3 .. pulling in a boat " }}{PARA 0 "" 0 "" {TEXT 309 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 364 "A rope attached to a boat is being pulled in at t he constant rate of 2.5 metres per sec. The rope is being drawn in fro m a point at the edge of the wharf which is 5 metres above the level o f the boat so that the boat moves towards the wharf. How fast is the \+ boat moving in the horizontal direction towards the wharf when 13 metr es of rope remains to be pulled in?" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{GLPLOT2D 619 270 270 {PLOTDATA 2 "69-%'CURVESG6$7$7$$\"\"!F)F(7$$ \"#@F)F(-%'COLOURG6&%$RGBGF)F)F)-F$6$7%7$$!\"%F)$\"\")F)7$F(F77$F($!\" $F)F--F$6$7%7$$!3w*************\\)!#=F77$$!3))**************HFC$\"3G** ***********p)!#<7$FEF7F--%)POLYGONSG6$7&7$$\"#:F)F(7$$\"$X\"!\"\"$\"\" \"F)7$$\"$&=FUFV7$$\"#=F)F(-F.6&F0$\"*++++\"!\")F(F(-FL6%7&F4F9F:7$F5F ;-%&COLORG6&F0$\"\"'FU$\"\"$FUFfo-%&STYLEG6#%,PATCHNOGRIDG-FL6$7U7$$\" \"#FU$\"#()FU7$$\"+1Ndg>!#5$\"+r=Fgp$\"+@mD&3 *Fjp7$$!*Cg'3KFgp$\"+jR;A\"*Fjp7$$!*TN5r)Fgp$\"+ENT_\"*Fjp7$$!+J]\"\\X \"Fgp$\"+e#Gb<*Fjp7$$!+HM4j?Fgp$\"+DO9\">*Fjp7$$!+/u/'o#Fgp$\"+kL,*>*F jp7$$!+)f_RJ$FgpF`t7$$!+tl!p$RFgpF[t7$$!+r\\3XXFgpFfs7$$!+ck*)G^FgpFas 7$$!+tR8zcFgpF\\s7$$!+Y*>r='Fgp$\"+AmD&3*Fjp7$$!+UJ%[k'Fgp$\"+_NFU!*Fj p7$$!+v\\3XqFgpF]r7$$!+-M`\"Q(FgpFhq7$$!+JC))[wFgpFcq7$$!+1e\"H%yFgp$ \"+V\\MC))Fjp7$$!+2NdgzFgp$\"+;mmi()Fjp7$$F\\oFU$\"+++++()Fjp7$$!+1Ndg zFgp$\"+$QLtj)Fjp7$Fev$\"+c]lv&)Fjp7$$!+IC))[wFgp$\"+Cx$f^)Fjp7$$!++M` \"Q(Fgp$\"+jJ7f%)Fjp7$$!+t\\3XqFgp$\"+ut51%)Fjp7$$!+QJ%[k'Fgp$\"+Zksd$ )Fjp7$$!+W*>r='Fgp$\"+yLu9$)Fjp7$$!+qR8zcFgp$\"+Pg$yF)Fjp7$$!+`k*)G^Fg p$\"+ukeZ#)Fjp7$$!+n\\3XXFgp$\"+UFgp$\"+%QLtj)Fjp7$ F`pF`w-F.6&F0F(FjnF(-F$6&7#FD-%'SYMBOLG6#%'CIRCLEGF--Fio6#%&POINTG-F$6 &F]^l-F_^l6#%(DIAMONDGF-Fb^l-F$6&F]^l-F_^l6#%&CROSSGF-Fb^l-F$6%7$7$$\" 3-+++++++;FC$\"3G************\\*)FI7$$\"3+++++++]9!#;FV-F.6&F0$\")#)eq kF\\o$\"))eqk\"F\\oF_`l-%*THICKNESSG6#Fap-F$6%7$7$$!3-+++++++jFC$\"3A* ***********>\"*FI7$$!3#)*************z%FIFj`lF[`lFa`l-FL6%7&F'F*7$F+F; F:-Fbo6&F0FfoFfo$\"\"*FUFho-F$6&7$7$FV$\"#WFU7$FV$\"#5FU7%7$$\"++++D6F jp$\"++++s7FjpF]bl7$$\"++++]()FgpFdblFhoF--F$6&7$7$FV$\"#cFU7$FV$\"$7* !\"#7%7$Fgbl$\"+++SQ))FjpF_cl7$FbblFeclFhoF--FL6&7&7$$!#GFU$\"#**FUF[d l7$$!+++++SFjpF^dlF`dl7%7$Fadl$\"++++S(*Fjp7$$!#VFUF^dl7$Fadl$\"++++15 F\\oFhoF[`l-F$6#7*F[dlF[dlF`dlFddlFgdlFjdlF`dlF[dl-F$6%7$7$$!35+++++++ ?FCFj`l7$$\"3%**************>\"FIFj`l-Fbo6&F0FfoFfoFfo-%*LINESTYLEGFc` l-F$6%7$7$F(FVFg_lFielF[fl-%%TEXTG6%7$$\"#6FU$\"\"&F)Q$5~m6\"F--Fbfl6% 7$$!#:FUF^dlQ(2.5~m/sFjflF--%+AXESLABELSG6%Q!FjflFdgl-%%FONTG6#%(DEFAU LTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$FhglFhgl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 "" {TEXT -1 62 "Let the horizo ntal distance between the boat and the wharf be " }{TEXT 323 1 "x" } {TEXT -1 86 " metres and the length of rope from the boat to the point at the edge of the wharf be " }{TEXT 324 1 "r" }{TEXT -1 9 " metres. \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 391 186 186 {PLOTDATA 2 "66-%'CURVESG6%7&7$$\"\"!F)$\"\"\"F)7$F($\"3A************>\"*!#<7$$ \"3+++++++]9!#;F*F'-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"F:F;-%*THI CKNESSG6#\"\"#-F$6$7%7$F($\"3(*************> " 0 "" {MPLTEXT 1 0 166 "p1 := plot(5*r/(2*sqrt(r^2-25)),r=0..20,0..10):\np2 := plot([[[ 0,2.5],[20,2.5]],[[5,0],[5,10]]],color=black,linestyle=3):\nplots[disp lay]([p1,p2],labels=[`r`,`dx/dt`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 416 302 302 {PLOTDATA 2 "6'-%'CURVESG6$7T7$$\"\"!F)F(7$%*undefinedGF+7$$\" 3i******\\1!>+&!#<$\"3?]u*z3([p!*!#;7$$\"3#)*****HSOa,&F/$\"3yv_xZ))*) )=$F27$$\"36+++d@(*G]F/$\"35?aX**pQKBF27$$\"3J+++5z]U]F/$\"3=**)3#>rVH >F27$$\"3k*****HmVg0&F/$\"3-+9;\")ft$o\"F27$$\"37+++q^6$3&F/$\"39%RF%p t8)Q\"F27$$\"3d*****\\n'=5^F/$\"3w\\!GIS5/@\"F27$$\"3e*****zoHV;&F/$\" 3[FAK<^z*)**F/7$$\"3T++++FZ=_F/$\"3LNcS()*f:t)F/7$$\"3O+++D(enK&F/$\"3 eP%GYY*[\\sF/7$$\"3J+++]Z/NaF/$\"3o1)**y*[=xjF/7$$\"3!)******\\?vVcF/$ \"3#4K.U\"\\:!R&F/7$$\"3;+++]$fC&eF/$\"3C.iJB!f.\"[F/7$$\"3$)*****Hd&) >/'F/$\"3g#4xAvGJX%F/7$$\"3Q+++'z6:B'F/$\"3S3_6l@&))=%F/7$$\"3U+++<=C# o'F/$\"3K%)*R.s\"QoPF/7$$\"3#)*****pEpS1(F/$\"3S=%e%*fV!RNF/7$$\"3@+++ j`A3vF/$\"3_+D^fy;^LF/7$$\"3v*****pwy8!zF/$\"3B2kNDAmGKF/7$$\"3e*****H OIFL)F/$\"3wRwi5s7DJF/7$$\"3&3++!)3zMu)F/$\"3So+)fbhu/$F/7$$\"3/+++#H_ ?<*F/$\"3#>0$*p]_?)HF/7$$\"3U+++!G;cc*F/$\"3s1SL[a]KHF/7$$\"3')******3 #G,***F/$\"3'*\\JXQJq()GF/7$$\"3'*******zw5V5F2$\"3m[lpawd[GF/7$$\"31+ ++%Q#\\\"3\"F2$\"3%\\\"[JgyS>GF/7$$\"3++++;*[H7\"F2$\"3o!GA)o#Q?z#F/7$ $\"3-+++qvxl6F2$\"3M!eG/2ouw#F/7$$\"3#******H0xw?\"F2$\"3kR8_m![mYGFF/7$$\"3/+++3'HKH\"F2$\"3EIsZdZ!3r#F/ 7$$\"32+++xanL8F2$\"3%RYu4E'o'p#F/7$$\"35+++v+'oP\"F2$\"3IFvBtK<$o#F/7 $$\"3%*******R<*fT\"F2$\"3q2C,*QL@n#F/7$$\"31+++&)Hxe9F2$\"3+0ow&e,7m# F/7$$\"3%******H!o-*\\\"F2$\"33wpJ8d'=l#F/7$$\"3&******4k.6a\"F2$\"3( \\zb5nqHk#F/7$$\"3%******>WTAe\"F2$\"3b+'[JVF]j#F/7$$\"32+++1*3`i\"F2$ \"34\\.PbqTFEF/7$$\"3;+++$*zym;F2$\"3XD,.EKp?EF/7$$\"3\"******4N1#4F2$\"37RmWZ]s*e#F/7$$\"33++ +/Uac>F2$\"3mh*Gc?jee#F/7$$\"#?F)$\"3w5;Z(*)))>e#F/-%'COLOURG6&%$RGBG$ \"#5!\"\"F(F(-F$6%7$7$F($\"3++++++++DF/7$FdzFc[l-Fiz6&F[[lF)F)F)-%*LIN ESTYLEG6#\"\"$-F$6%7$7$$\"\"&F)F(7$F`\\l$F][lF)Ff[lFh[l-%+AXESLABELSG6 %%\"rG%&dx/dtG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fdz;F(Fc\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 413 "When the length of \+ rope remaining to be drawn in is large, the speed of the boat is appro ximately equal to the rate at which the rope is drawn in. However, wh en the boat nears the wharf, as the remaining length of rope approache s 5 metres, the speed of the boat increases and approaches infinity. B ecause this is impossible in a practical situation, the rate at which \+ the rope is drawn in would have to decrease." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 320 "If the question is reworded to say simply that, at the instant when the length of rope remaining to \+ be pulled in is 13 metres, the rate at which it is being pulled in is \+ 2.5 metres per sec, then this would allow for the possibility that thi s rate could reduce. This does not alter the way that the question is \+ answered. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 159 "The length of a side of a cube is increa sing at a rate of 6 cm per sec. Find the rate of increase of the volum e of the cube when the length of a side is 9 cm. " }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 20 " 1458 cu cm pe r sec " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 156 "Air is bein g blown into a spherical balloon at a constant rate of 1000 cubic cm p er sec. What is the rate of change of the radius when the radius is 10 cm? " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 37 ": The volume of a sphere with radius " }{TEXT 320 1 "r" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "4/3" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Pi*r^3" "6#*&%#PiG\"\"\"*$%\"rG\"\"$F%" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "5/2/Pi" "6#*(\"\"&\"\"\"\"\"#!\"\"%#PiGF'" } {TEXT -1 1 " " }{TEXT 326 1 "~" }{TEXT -1 26 " 0.7957747152 cm per sec . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "1000/(4*Pi*100);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&\"\"\"\"\"#!\"\"%#PiGF(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_rudz!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 268 "At a given instant the radii of two conc entric circles are 8 cm and 12 cm. The radius of the outer circle incr eases at a rate of 1 cm per sec and that of the inner circle at a rate of 2 cm per sec. Find the rate of change of the area enclosed between the two circles. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 36 "The area is decreasing at a rate of " }{XPPEDIT 18 0 "8*Pi" "6#*&\"\")\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 327 1 "~" }{TEXT -1 32 " 25.13274123 square cm per sec. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "8*Pi;\nevalf (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\")\"\"\"%#PiGF&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BTF8D!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 150 "A container which has the shape of \+ a hollow right circular cone with its vertex downwards is held beneath a tap leaking at a rate of 2 cu cm per sec. " }}{PARA 0 "" 0 "" {TEXT -1 70 "The height of the conical container is 18 cm and its radi us is 12 cm. " }}{PARA 0 "" 0 "" {TEXT -1 103 "Find the rate at which \+ the water level in the container is rising when the depth of the water is 6 cm. " }}{PARA 0 "" 0 "" {TEXT -1 55 "The volume of a right circu lar cone with a base radius " }{TEXT 321 1 "r" }{TEXT -1 12 " and heig ht " }{TEXT 322 1 "h" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "1/3" "6#*&\" \"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*r^2*h" "6#*(%#PiG \"\"\"*$%\"rG\"\"#F%%\"hGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(8*Pi )" "6#*&\"\"\"F$*&\"\")F$%#PiGF$!\"\"" }{TEXT -1 1 " " }{TEXT 328 1 "~ " }{TEXT -1 26 " 0.03978873576 cm per sec " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "1/(8*Pi);\nevalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*&\"\")F%%#PiGF%!\" \"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wN()yR!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 254 "A weather balloon le aves the ground 275 metres from an observer and rises vertically at a \+ rate of 12 metres per sec. How fast is the length of the line of sight from the observer to the balloon increasing when the balloon is 450 m etres above the ground. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "216/sqrt(445)" "6#*&\" $;#\"\"\"-%%sqrtG6#\"$X%!\"\"" }{TEXT -1 1 " " }{TEXT 329 1 "~" } {TEXT -1 28 " 10.23938200 metres per sec " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "hh := 450;\nkk := 275 ;\nxx := sqrt(hh^2+kk^2);\n18/sqrt(445)*12;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhG\"$]%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# kkG\"$v#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"#D\"\"\"\"$X%# F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"$;#\"\"\"\"$X%!\"\" F'#F&\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++#QR-\"!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 204 "A man 1.8 metres ta ll approaches a street light 4.5 metres above the ground at a rate of \+ 1.5 metres per sec. How fast is the end of the man's shadow moving whe n he is 3 metres from the base of the light? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 19 "2.5 metres per sec " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 29 "code for oil slick animation " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 407 "ocean := plots[polygonplot]([[-1.5,-1.1],[-1.5,1.1],\n [1.5,1 .1],[1.5,-1.1]],color=COLOR(RGB,.75,.75,.9)):\nfrms := NULL:\nnumframe s := 30:\nfor i to numframes do\n rr := evalf(sqrt(i/numframes));\n \+ oil := plottools[disk]([0,0],rr,color=COLOR(RGB,.75,.5,.5));\n frm s := frms,plots[display]([oil,ocean]);\nend do:\nplots[display]([frms] ,insequence=true,\n scaling=constrained,axes=none,style=patchnogrid );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 7 "circle " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "p1 := plottools[circle]([0,0],1,color=COLOR(RGB,.55,.2,.2)):\np 2 := plottools[arrow]([0,0],[sqrt(3)/2,1/2],0,.1,.1,\n arrow,c olor=COLOR(RGB,.55,.2,.2)):\np3 := plot([[[0,0]]$3],style=point,\n \+ symbol=[circle,cross,diamond],color=COLOR(RGB,.55,.2,.2)):\nt1 : = plots[textplot]([.44,.18,`r`],font=[TIMES,ITALIC,12]):\nplots[displa y]([p1,p2,p3,t1],scaling=constrained,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "code for sawdust pile animation " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "nn := 40:\ntb := plots[tubeplot]([0,-s,1.5],s=-1..-.02,radius=0.0 7,color=COLOR(RGB,.65,.65,.9)):\nfrms := NULL:\nfor k from 1 to 40 do \n t := evalf((k/40)^(1/3));\n u := k*0.8;\n ss := plots[spacecu rve]([.015*cos(s*80+u),.015*sin(s*80+u),s],s=t..1.5,\n thickness= 2,color=COLOR(RGB,.6,.4,.1)):\n aa := [0,0,t]: \n b := [seq(eval f([t*cos(2*Pi*i/nn),t*sin(2*Pi*i/nn),0]),i=1..nn+1)]:\n sd := POLYGO NS([seq(b[i],i=1..nn)],seq([b[i],b[i+1],aa],i=1..nn)):\n if k<>1 and k<>40 then \n frms := frms,plots[display]([tb,ss,sd]);\n else \n frms := frms,plots[display]([tb,sd]);\n end if;\nend do:\npl ots[display]([frms],style=patchnogrid,color=COLOR(RGB,.6,.4,.1),\n l ightmodel=light2,insequence=true,orientation=[-170,67]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 257 "" 0 "" {TEXT -1 27 "code for projection of cone" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "p1 := plot([cos(t),.2*sin(t),t=-Pi..0],color=brown):\np2 := plo t([cos(t),.2*sin(t),t=0..Pi],color=brown,linestyle=2):\np3 := plot([[[ -1,0],[0,1],[1,0],[-1,0]],[[0,0],[0,1]],\n [[0,.07],[.07,.07],[.07, 0]]],color=[brown$2,black]):\nt1 := plots[textplot]([[-0.05,.5,`h`],[. 5,-.05,`r`]],color=black):\nplots[display]([p1,p2,p3,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 36 "code for boat pic ture and triangle " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1074 "p1 := plot([[[0,0],[21,0]],[[-4,8],[0,8], [0,-3]],\n [[-.85,8],[-.3,8.7],[-.3,8]]],color=black):\np2 := plot s[polygonplot]([[15,0],[14.5,1],[18.5,1],[18,0]],\n color=red):\np3 \+ := plots[polygonplot]([[-4,8],[0,8],[0,-3],[-4,-3]],\n color=COLOR(R GB,.6,.3,.3),style=patchnogrid):\np4 := plottools[disk]([-.3,8.7],.5,c olor=green):\np5 := plot([[[-.3,8.7]]$3],style=point,color=black,\n \+ symbol=[circle,diamond,cross]):\np6 := plot([[[.16,8.95],[14.5,1]],[ [-.63,9.12],[-4.8,9.12]]],\n color=brown,thickness=2):\np7 := plot s[polygonplot]([[0,0],[21,0],[21,-3],[0,-3]],\n color=COLOR (RGB,.3,.3,.9),style=patchnogrid):\np8 := plottools[arrow]([1,4.4],[1, 1],0,.25,.08,arrow,color=black):\np9 := plottools[arrow]([1,5.6],[1,9. 12],0,.25,.08,arrow,color=black):\np10 := plottools[arrow]([-2.8,9.9], [-4.3,9.9],0,.32,.2,\n color=brown):\np11 := plot([[[-.2,9.12],[1.2, 9.12]],\n [[0,1],[14.5,1]]],color=COLOR(RGB,.3,.3,.3),linestyle=2) :\nt1 := plots[textplot]([[1.1,5,`5 m`],[-1.5,9.9,`2.5 m/s`]],color=bl ack):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,t1],axes=non e);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1013 "p1 := plot([[0,1],[0,9.1 2],[14.5,1],[0,1]],color=brown,\n thickness=2):\np2 := plot([[0, 1.72],[.54,1.72],[.54,1]],color=black):\np3 := plot([[[-1,1],[0,1]],[[ -1,9.12],[0,9.12]],\n [[0,-.5],[0,1]],[[14.5,-.5],[14.5,1]],\n [ [0,9.12],[.3,10.3]],[[14.5,1],[14.8,2.18]]],\n color=black,li nestyle=2):\np4 := plottools[arrow]([-.5,5.75],[-.5,9.12],0,.25,.08,\n arrow,color=blue):\np5 := plottools[arrow]([-.5,4.35],[-.5,1],0,. 25,.08,\n arrow,color=blue):\np6 := plottools[arrow]([7,.25],[0,.2 5],0,.3,.03,\n arrow,color=blue):\np7 := plottools[arrow]([8,.25], [14.5,.25],0,.3,.03,\n arrow,color=blue):\np8 := plottools[arrow]( [7,5.94],[.15,9.71],0,.3,.03,\n arrow,color=blue):\np9 := plottool s[arrow]([8,5.36],[14.65,1.59],0,.3,.03,\n arrow,color=blue):\nt1 \+ := plots[textplot]([[-.5,5.06,`5`]],\n color=black,font=[HELVETI CA,10]):\nt2 := plots[textplot]([[7.5,.33,`x`],[7.6,5.8,`r`]],\n \+ color=black,font=[TIMES,ITALIC,10]):\nplots[display]([p1,p2,p3,p4,p 5,p6,p7,p8,p9,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }