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0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Volumes of solids" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 29.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Some standard solids" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "A cube" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "plots[pol ygonplot3d]([[[0,0,0],[1,0,0],[1,1,0],[0,1,0]],\n[[0,0,1],[1,0,1],[1,1 ,1],[0,1,1]],\n[[0,0,0],[0,1,0],[0,1,1],[0,0,1]],\n[[1,0,0],[1,1,0],[1 ,1,1],[1,0,1]],\n[[0,0,0],[1,0,0],[1,0,1],[0,0,1]],\n[[0,1,0],[1,1,0], [1,1,1],[0,1,1]]],\n lightmodel=light4,scaling=constrained,style=patc hnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 16 "volume of a cube" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "s^3;" "6#*$%\"sG\"\"$" }{TEXT -1 48 ", where s is the l ength of one side of the cube." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "A cylinder" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "h := Pi/25: centre := 'centre':\nc ircle := seq(evalf([cos(h*k),sin(h*k),centre]),k=0..50):\ncentre := 3: top := circle:\ncentre := 0: bottom := circle:\np := PLOT3D(MESH([[to p],[bottom]]),POLYGONS([top],[bottom])):\nplots[display](p,lightmodel= light2,scaling=constrained,style=patchnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "A cylinder can also be dr awn by using the procedure " }{TEXT 0 8 "cylinder" }{TEXT -1 10 " from the " }{TEXT 0 9 "plottools" }{TEXT -1 95 " package.\nYou can specify the coordinates of the centre of the base the radius and the height. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plots[display](plottools[cylinder]([0,0,0],1,3),scaling=constr ained,\n style=patch,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 20 "volume of a cylinde r" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Pi*r^2*h;" "6#*(%#PiG\"\"\"*$% \"rG\"\"#F%%\"hGF%" }{TEXT -1 8 ", where " }{TEXT 276 1 "r" }{TEXT -1 31 " is the radius of the base and " }{TEXT 277 1 "h" }{TEXT -1 29 " i s the perpendicular height." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "A cone" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 183 "h := Pi/25:\ncircle := seq(evalf([cos(h*k), sin(h*k),0]),k=0..50):\nplots[polygonplot3d]([[[0,0,3],circle],[[0,0,0 ],circle]],\n lightmodel=light2,scaling=constrained,style=patchnogri d);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "S trictly speaking this is a " }{TEXT 259 19 "right circular cone" } {TEXT -1 134 ", because the base 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^q7&F[\\qF]\\q7%Fj^lFbjmF]qFj^q7&F]\\qF_\\q7%F^_lFfjmFjnF\\_q7&F_\\qFa \\q7%Fd_lFjjmF_rF^_q7&Fa\\qFc\\q7%Fh_lF^[nFjrF`_q7&Fc\\qFe\\q7%F^`lF]q FesFb_q7&Fe\\qFg\\q7%Fb`lFd[nF`tFd_q7&Fg\\qFi\\q7%Fh`lFh[nF[uFf_q7&Fi \\qF[]q7%F\\alF\\\\nFcuFh_q7&F[]qF]]q7%FbalF`\\nF\\vFj_q7&F]]qF_]q7%Ff alFd\\nFgvF\\`q7&F_]qFa]q7%F\\blFh\\nFbwF^`q7&Fa]qFc]q7%F`blF\\]nF]xF` `q7&Fc]qFe]q7%FfblF`]nFfxFb`q7&Fe]qFg]q7%FjblFd]nFayFd`q7&Fg]qFi]q7%F` clFbpF\\zFf`q7&Fi]qF[^q7%FdclFj]nFgzFh`q7&F[^qF]^q7%FjclF^^nFb[lFj`q7& F]^qF_^q7%F^dlFb^nFj[lF\\aq7&F_^qFa^q7%FddlFf^nFe\\lF^aq7&Fa^qFc^q7%Fh dlFj^nF`]lF`aq7&Fc^qFZFenFbaq7&FhnFf^q7%F_oFicmF\\oFhn7&Ff^qFh^q7%FjoF ]dmFgoFeaq7&Fh^qFj^q7%FepFadmFbpFgaq7&Fj^qF\\_q7%F`qFedmF]qFiaq7&F\\_q F^_q7%FiqFidmFjnF[bq7&F^_qF`_q7%FbrF]emF_rF]bq7&F`_qFb_q7%F]sFaemFjrF_ bq7&Fb_qFd_q7%FhsFeemFesFabq7&Fd_qFf_q7%FctFiemF`tFcbq7&Ff_qFh_q7%F]uF ]fmF[uFebq7&Fh_qFj_q7%FfuFafmFcuFgbq7&Fj_qF\\`q7%F_vFefmF\\vFibq7&F\\` qF^`q7%FjvFifmFgvF[cq7&F^`qF``q7%FewF]gmFbwF]cq7&F``qFb`q7%F`xFagmF]xF _cq7&Fb`qFd`q7%FixFegmFfxFacq7&Fd`qFf`q7%FdyFigmFayFccq7&Ff`qFh`q7%F_z F]hmF\\zFecq7&Fh`qFj`q7%FjzFahmFgzFgcq7&Fj`qF\\aq7%Fd[lFehmFb[lFicq7&F \\aqF^aq7%F]\\lFihmFj[lF[dq7&F^aqF`aq7%Fh\\lF]imFe\\lF]dq7&F`aqFbaq7%F c]lFaimF`]lF_dq7&FbaqFenFfnFadq7&FhnFeaqFinFhn7&FeaqFgaqFdoFin7&FgaqFi aqF_pFdo7&FiaqF[bqFjpF_p7&F[bqF]bqFeqFjp7&F]bqF_bqF^rFeq7&F_bqFabqFgrF ^r7&FabqFcbqFbsFgr7&FcbqFebqF]tFbs7&FebqFgbqFhtF]t7&FgbqFibqFbuFht7&Fi bqF[cqF[vFbu7&F[cqF]cqFdvF[v7&F]cqF_cqF_wFdv7&F_cqFacqFjwF_w7&FacqFccq FexFjw7&FccqFecqF^yFex7&FecqFgcqFiyF^y7&FgcqFicqFdzFiy7&FicqF[dqF_[lFd z7&F[dqF]dqFi[lF_[l7&F]dqF_dqFb\\lFi[l7&F_dqFadqF]]lFb\\l7&FadqFfnF'F] ]l-%&STYLEG6#%&PATCHG-%*AXESSTYLEG6#%%NONEG-%+PROJECTIONG6%$\"#gF*$!#! )F*F," 1 2 0 1 10 0 2 1 1 1 2 1.000000 -80.000000 60.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 259 16 "volume of a cone" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Pi*r^2*h/3;" "6#**%#PiG\"\"\"*$%\"rG\"\"#F%%\"hGF%\"\"$!\"\"" } {TEXT -1 8 ", where " }{TEXT 278 1 "r" }{TEXT -1 31 " is the radius of the base and " }{TEXT 279 1 "h" }{TEXT -1 15 " is the height." }} {PARA 0 "" 0 "" {TEXT -1 57 "An artificial ice-cream cone (from the Ma ple help pages)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 213 "icecream := plottools[cone]([0,0,-1.5],0.7,1. 5,color=gold),plottools[sphere]([0,0,0.1],0.6,color=pink):\nplots[disp lay](icecream,scaling=constrained,style=patchnogrid,orientation=[32,74 ],\n lightmodel=light4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 8 "A sphere" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "We can draw a sphere by means of a tubeplot with var ing radius " }{XPPEDIT 18 0 "sin*t;" "6#*&%$sinG\"\"\"%\"tGF%" }{TEXT -1 11 " at height " }{XPPEDIT 18 0 "cos*t;" "6#*&%$cosG\"\"\"%\"tGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "plots[tubeplot]([0,0,cos(t)],t=0..Pi,radius=sin (t),tubepoints=50,\n lightmodel=light1,scaling=constrained,style=pat chnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "A sphere can also be drawn by using two parameters s and t whe re the parameter s draws a circle of radius " }{XPPEDIT 18 0 "sin*t;" "6#*&%$sinG\"\"\"%\"tGF%" }{TEXT -1 11 " at height " }{XPPEDIT 18 0 "c os*t;" "6#*&%$cosG\"\"\"%\"tGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot3d([sin (t)*cos(s),sin(t)*sin(s),cos(t)],t=0..Pi,s=0..2*Pi,\n lightmodel=lig ht2,scaling=constrained,style=patchnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can calculate our own mesh o f points to plot a sphere." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "h := Pi/25:\nc := seq(evalf(cos(h* k)),k=0..50):\ns := seq(evalf(sin(h*k)),k=0..50):\nm := [seq([seq([s[j ]*c[i],s[j]*s[i],c[j]],i=1..51)],j=1..26)]:\np := PLOT3D(MESH(m)):\npl ots[display](p,lightmodel=light3,scaling=constrained,style=patchnogrid );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "A \+ sphere can also be drawn by using the procedure " }{TEXT 0 6 "sphere" }{TEXT -1 10 " from the " }{TEXT 0 9 "plottools" }{TEXT -1 10 " packag e.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "with(plottools):\npl ots[display](sphere(),scaling=constrained,style=patch,lightmodel=light 4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Y ou can specify the coordinates of the centre of the base the radius an d the height." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "plots[display]([sphere([0,0,0],3),sphere([0,0,5 ],1)],scaling=constrained,\n lightmodel=light3,style= patchnogrid,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 18 "volume of a sphere" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "4*Pi*r ^3/3;" "6#**\"\"%\"\"\"%#PiGF%%\"rG\"\"$F(!\"\"" }{TEXT -1 8 ", where \+ " }{TEXT 280 1 "r" }{TEXT -1 23 " is the radius and its " }{TEXT 259 12 "surface area" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "4*Pi*r^2;" "6#*( \"\"%\"\"\"%#PiGF%%\"rG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "A torus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 45 "Torus is a fancy name for a ring or doug hnut." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "plots[tubeplot]([cos(t),sin(t),0],t=0..2*Pi,radius=0 .35,numpoints=50,tubepoints=50,\n lightmodel=light2,scaling=constrai ned,style=patchnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 49 "A torus can also be drawn by using the procedure \+ " }{TEXT 0 5 "torus" }{TEXT -1 10 " from the " }{TEXT 0 9 "plottools" }{TEXT -1 10 " package.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plots[display](plottools[torus](),scaling=constrained,style=patch);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 259 17 "volume of a torus" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "4 *Pi^2*R^2*r;" "6#**\"\"%\"\"\"*$%#PiG\"\"#F%%\"RGF(%\"rGF%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 67 " is the radiu s of the small circular cross-section of the ring and " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 107 " is the radius of the circle which joi ns all the centres of the circular cross sections of the ring itself. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "A triangular prism " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "bottom := plots[polygonplot3d]([[0,0,0],[1,0,0],[0,1,0]]):\nt op := plots[polygonplot3d]([[0,0,3],[1,0,3],[0,1,3]]):\nsides := plots [polygonplot3d]([[[0,0,0],[0,1,0],[0,1,3],[0,0,3]],\n [[0,0,0],[1,0, 0],[1,0,3],[0,0,3]],[[0,1,0],[1,0,0],[1,0,3],[0,1,3]]]):\nplots[displa y3d]([top,bottom,sides],lightmodel=light4,style=patchnogrid);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The volum e of a " }{TEXT 259 16 "triangular prism" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "A*h;" "6#*&%\"AG\"\"\"%\"hGF%" }{TEXT -1 9 ", where " }{TEXT 281 1 "A" }{TEXT -1 29 " is the area of the base and " }{TEXT 282 1 "h" }{TEXT -1 15 " is the height." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A \+ procedure for constructing a prism: " }{TEXT 0 5 "prism" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 5 "pri sm" }{TEXT -1 43 " is any solid with a uniform cross section." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 12 "prism: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 18 " prism(c, ...)" }}{PARA 0 "" 0 "" {TEXT -1 20 " prism(c, n ...)" }}{PARA 0 "" 0 "" {TEXT -1 24 " prism(c, n, r, ...)" }}{PARA 0 "" 0 "" {TEXT -1 22 " prism(c, n, r, h)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Paramet ers:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 19 "c - " }{TEXT -1 63 "centre of t he base of the prism (optional, default is [0,0,0])" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 23 1 " " }{TEXT -1 3 " " }{TEXT 262 19 "n \+ - " }{TEXT -1 53 "number of sides for the base (optional, \+ default is 6)" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 19 "r \+ - " }{TEXT -1 81 "radius of the circumscribing circle for the base polygon (optional, default is 1)" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 19 "h - " }{TEXT -1 46 "height o f the prism (optional, default is 3*r)" }}{PARA 0 "" 0 "" {TEXT -1 11 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 219 "The routine p rism creates a three-dimensional plot data object which when displayed , is a prism with n sides and height h. The base is a regular polygon \+ centered at c. The resulting prism object is capped on both ends. " }} {PARA 15 "" 0 "" {TEXT -1 140 "A call to prism produces a plot data ob ject, which can be used in a PLOT3D data structure, or displayed using the function plots[display]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ e open the subsection, place the cursor anywhere after the prompt [ > \+ and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "prism: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "prism" {MPLTEXT 1 0 873 "prism \+ := proc(centre::list(realcons),numsides::posint,radius::realcons,heigh t::realcons)\n local i,p,b,t,c,n,h,r;\n c := [0,0,0];\n n := 6; \n r := 1.;\n if nargs>=1 then\n if nops(centre)=3 then\n \+ c := evalf(centre);\n else\n error \"1st argument mu st be a list of 3 real constants\"\n end if;\n end if;\n if n args>=2 then\n if numsides>=3 then\n n := numsides;\n \+ else\n error \"number of sides must be at least 3\"\n e nd if;\n end if;\n if nargs>=3 then r := evalf(abs(radius)) end if ;\n h := r*3;\n if nargs>=4 then h := evalf(height) end if;\n p \+ := [seq(evalf([r*cos(2*Pi*i/n)+c[1],r*sin(2*Pi*i/n)+c[2]]),i=1..n+1)]; \n b := [seq([p[i,1],p[i,2],c[3]],i=1..n+1)]:\n t := [seq([p[i,1], p[i,2],c[3]+h],i=1..n+1)]:\n POLYGONS([seq(b[i],i=1..n)],seq([b[i],b [i+1],t[i+1],t[i]],i=1..n),[seq(t[i],i=1..n)]);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next sectio n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "prism" }{TEXT -1 10 ": examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plots[displa y](prism(),scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "Centre of base" }{XPPEDIT 18 0 " ``(1,1,0 )" "6#-%!G6%\"\"\"F&\"\"!" }{TEXT -1 61 ", 7 sides, radius of circumbs cribing circle = 1, height = 10." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plots[display](prism([1,1,0] ,7,2,10),scaling=constrained,axes=normal);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plots[display3d]( [prism([0,0,0],4,1,3),prism([3,3,0],6,2,6),\n prism([10,10,0],8,3,8) ],scaling=constrained,orientation=[0,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "A procedure for constructing a pyramid: \+ " }{TEXT 0 7 "pyramid" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 7 "pyramid" }{TEXT -1 87 " is solid form ed by joining all the vertices of a base polygon to a single point - t he " }{TEXT 258 4 "apex" }{TEXT -1 16 " of the pyramid." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "pyramid: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 18 "Calling Sequence:\n" }}{PARA 0 "" 0 " " {TEXT -1 22 " pyramid( c, ... )" }}{PARA 0 "" 0 "" {TEXT -1 24 " pyramid( c, n ... )" }}{PARA 0 "" 0 "" {TEXT -1 28 " pyramid( c, n, r, ... )" }}{PARA 0 "" 0 "" {TEXT -1 53 " pyramid( c, n, r, h ) or pyramid( c, n, r, a )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 19 "c \+ - " }{TEXT -1 65 "centre of the base of the pyramid (optiona l, default is [0,0,0])" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 19 "n - " }{TEXT -1 53 "number of sides for the base (optional, default is 6)" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {TEXT 23 19 "r - " }{TEXT -1 81 "radius of the circumsc ribing circle for the base polygon (optional, default is 1)" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 19 "h - " } {TEXT -1 46 "height of the prism (optional, default is 3*r)" }}{PARA 0 "" 0 "" {TEXT 259 2 "or" }{TEXT -1 29 " a \+ " }{TEXT 23 5 " - " }{TEXT -1 29 "the 3 coordinates of the apex" }} {PARA 0 "" 0 "" {TEXT -1 11 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 " " {TEXT -1 282 "The routine pyramid creates a three-dimensional plot d ata object which when displayed, is a pyramid with n sides and height \+ h. The base is a regular polygon centered at c. The apex is directly a bove the centre of the base unless the coordinates are specified inste ad of the height. " }}{PARA 15 "" 0 "" {TEXT -1 140 "A call to prism p roduces a plot data object, which can be used in a PLOT3D data structu re, or displayed using the function plots[display]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To ma ke the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the s ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "pyramid: implementa tion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "pyramid " {MPLTEXT 1 0 1005 "pyramid := proc(centre::list(realcons),numsides:: posint,radius::realcons,apex)\n local i,b,a,c,n,r;\n c := [0,0,0]; \n n := 5;\n r := 1.;\n if nargs>=1 then\n if nops(centre)= 3 then\n c := evalf(centre);\n else\n error \"1st argument must be a list of 3 real constants\"\n end if;\n end \+ if;\n if nargs>=2 then\n if numsides>=3 then\n n := num sides;\n else\n error \"number of sides must be at least \+ 3\"\n end if;\n end if;\n if nargs>=3 then r := evalf(abs(rad ius)) end if;\n a := [c[1],c[2],3*r];\n if nargs>=4 then\n if type(apex,list(realcons)) and nops(apex)=3 then\n a := evalf( apex);\n elif type(apex,realcons) then\n a := [c[1],c[2], evalf(apex)];\n else\n error \"3rd argument must be a rea l constant or a list of 3 real constants\"\n end if;\n end if; \n b := [seq(evalf([r*cos(2*Pi*i/n)+c[1],r*sin(2*Pi*i/n)+c[2],c[3]]) ,i=1..n+1)];\n POLYGONS([seq(b[i],i=1..n)],seq([b[i],b[i+1],a],i=1.. n));\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 39 "Examples ar e given in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 7 "pyramid" }{TEXT -1 10 ": \+ examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plots[display](pyramid(),scaling=constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Centre of base" }{XPPEDIT 18 0 "``(1,1,0)" "6#-%!G6%\"\"\"F&\"\"!" }{TEXT -1 60 ", 7 sides, radius of circumbscribing circle = 2, height = 5." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plots[display](pyramid([1,1,0],7,2,5),axes=normal,orientation=[25, 65]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Centre of base" }{XPPEDIT 18 0 "``(1,1,0)" "6#-%!G6%\"\"\"F&\"\"!" } {TEXT -1 56 ", 7 sides, radius of circumbscribing circle = 2, apex at " }{XPPEDIT 18 0 "``(5,5,5)" "6#-%!G6%\"\"&F&F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plots[display](pyramid([1,1,0],7,2,[5,5,5]),axes=normal,orientatio n=[25,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "plots[display3d]([pyramid([0,0,0],4,1),pyramid([3 ,3,0],6,2),pyramid([10,10,0],8,3)],\n scaling=constrained,orientati on=[0,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The vol ume of a prism" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 259 5 "prism" }{TEXT -1 43 " is any solid with a uniform cross section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 17 "volume of a pr ism" }{TEXT -1 13 " is given by " }{XPPEDIT 18 0 "A*h;" "6#*&%\"AG\"\" \"%\"hGF%" }{TEXT -1 8 ", where " }{TEXT 274 1 "A" }{TEXT -1 48 " is t he area of the base (or cross-section) and " }{TEXT 275 1 "h" }{TEXT -1 15 " is the height." }}{PARA 0 "" 0 "" {TEXT -1 369 "To see this im agine that the base of the prism is subdivided into many small squares . Then construct prisms with a square cross-section on each of these s mall squares as base and having height equal to the height of the orig inal prism. Since the volume of each of the square prisms is given by \+ the area of its base times its height, the same is true of the large p rism." }}{PARA 0 "" 0 "" {TEXT -1 40 "The next picture requires the pr ocedure " }{TEXT 0 5 "prism" }{TEXT -1 5 " . . " }{HYPERLNK 17 "prism " 1 "" "prism" }{TEXT -1 12 " from above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "p1 := plots[display] (prism([0,0,0],7,5,15),style=wireframe,thickness=2):\np2 := plots[disp lay](prism([2,0,0],4,0.5,15)):\nplots[display]([p1,p2],scaling=constra ined,orientation=[45,55]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%)POLYGONSG6-7)7%$\"+4!\\u6$!\"*$\"+7u:4RF*$\"\"!F.7%$ !+oYg76F*$\"+i&RY([F*F-7%$!+RV%[]%F*$\"+'p=%p@F*F-7%F5$!+'p=%p@F*F-7%F 0$!+i&RY([F*F-7%F($!+7u:4RF*F-7%$\"\"&F.F-F-7&F'F/7%F0F2$\"#:F.7%F(F+F G7&F/F47%F5F7FGFF7&F4F97%F5F:FGFK7&F9F<7%F0F=FGFM7&F " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "The volu me of a cylinder" }}{PARA 0 "" 0 "" {TEXT -1 72 "Since a cylinder is a prism with a circular cross-section its volume is " }{XPPEDIT 18 0 "P i*r^2*h;" "6#*(%#PiG\"\"\"*$%\"rG\"\"#F%%\"hGF%" }{TEXT -1 8 ", where \+ " }{TEXT 285 1 "r" }{TEXT -1 19 " is the radius and " }{TEXT 286 1 "h " }{TEXT -1 15 " is the height." }}{PARA 0 "" 0 "" {TEXT -1 40 "The ne xt picture requires the procedure " }{TEXT 0 5 "prism" }{TEXT -1 5 " . . " }{HYPERLNK 17 "prism" 1 "" "prism" }{TEXT -1 12 " from above." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "p1 := plots[display](plottools[cylinder]([0,0,0],4,15,strips=50), style=wireframe):\np2 := plots[display](prism([2,0,0],4,0.5,15)):\nplo ts[display]([p1,p2],scaling=constrained,orientation=[30,60]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "The volume of a pyramid 1 . . sheering transformation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 82 "The first step in determining \+ a formula for the volume of a prism is to note that " }{TEXT 259 76 "t he volume of all prisms with the same base and the same height are the same" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "This can be expl ained with reference to the idea of a " }{TEXT 259 23 "sheering transf ormation" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 139 "Imagine that the prism is sliced into many very thin layers parallel to the base and that these thin layers can all slide over one another." }}{PARA 15 "" 0 "" {TEXT -1 190 "All ow the layers to slide so that the corners all stay in a straight line . The distance of the apex above the plane of the base does not change and the volume of the pyramid does not change." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Try the following two ani mations. They require the procedure " }{TEXT 0 7 "pyramid" }{TEXT -1 5 " . . " }{HYPERLNK 17 "pyramid" 1 "" "pyramid" }{TEXT -1 12 " from a bove." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "The first animation demonstrates the sheering of a prism with a squar e base so that the apex follows a straight line parallel to the plane \+ of the base." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "n := 4: k := 'k':\nplots[display]([seq(pyramid([0 ,0,0],n,3,[0,k*0.1,5]),k=-40..40)],\n scaling=constrained,orientatio n=[0,65],insequence=true,\n title=\"sheering animation for a pris m\");" }}{PARA 13 "" 1 "" {GLPLOT3D 486 486 486 {PLOTDATA 3 "6&-%(ANIM ATEG6]p7#-%)POLYGONSG6'7&7%$\"\"!F-$\"\"$F-F,7%$!\"$F-F,F,7%F,F1F,7%F. 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At this stage two of its faces are perpendicular to the base and are right-angled triangles. " }}{PARA 0 "" 0 "" {TEXT -1 130 "It can then be sheered parallel to one of these right-angled tria ngle faces so that the base also becomes a right-angled triangle." }} {PARA 0 "" 0 "" {TEXT -1 115 "This first animation results in a pyrami d with three mutually perpendicular faces which are right-angled trian gles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 437 "v1:=[0,0,0]: v2:=[7,2,0]: v3:=[0,6,0]: v5:=[0,0,5]: \nv4:=[0.1*(20-k),0.15*(20-k),5]: v6:=[7,0.1*(20-k),0]:\nplots[display ](seq(plots[polygonplot3d]([[v1,v2,v3],[v1,v2,v4],\n \+ [v1,v2,v4],[v2,v3,v4]]),k=0..20),\nseq(plots[polygonplot3d]([[v1,v6,v 3],[v1,v6,v5],\n [v1,v6,v5],[v6,v3,v5]]),k=0..20),\n lightmodel=light3,scaling=constrained,style=patchnogrid,\n orient ation=[-5,80],axes=normal,insequence=true);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 209 "This second animation results \+ in a pyramid with three faces which are right-angled triangles which a re not mutually perpendicular, but nevertheless the pyramid can also b e obtained as a dissection of a cuboid." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "v1:=[0,0,0]: v2:=[7,2, 0]: v3:=[0,6,0]: v5:=[0,0,5]:\nv4:=[0.1*(20-k),0.15*(20-k),5]: v6:=[7, 2+0.2*k,0]:\nplots[display](seq(plots[polygonplot3d]([[v1,v2,v3],[v1,v 2,v4],\n [v1,v2,v4],[v2,v3,v4]]),k=0..20),\nseq(plot s[polygonplot3d]([[v1,v6,v3],[v1,v6,v5],\n [v1,v6,v5 ],[v6,v3,v5]]),k=0..20),\n lightmodel=light3,scaling=constrained,sty le=patchnogrid,\n orientation=[-5,80],axes=normal,insequence=true); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "The volume of a pyramid 2 . . dissection of a cuboid" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We start with a cuboid." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "A1 := [4,0,4]: B1 := [0,0,4]: C1 := [0,0,0]: D1 := [4,0,0]:\nA2 : = [4,5,4]: B2 := [0,5,4]: C2 := [0,5,0]: D2 := [4,5,0]:\nCuboid:=[[C1, D1,D2,C2],[B1,A1,A2,B2],[C1,C2,B2,B1],\n [D1,D2,A2,A1],[C1,D1,A1,B1] ,[C2,D2,A2,B2]]:\nplots[polygonplot3d](Cuboid,style=patch,lightmodel=l ight3,\n scaling=constrained,orientation=[40,60]);" }}{PARA 13 " " 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6(7&7%$\"\"!F) F(F(7%$\"\"%F)F(F(7%F+$\"\"&F)F(7%F(F.F(7&7%F(F(F+7%F+F(F+7%F+F.F+7%F( F.F+7&F'F0F5F27&F*F-F4F37&F'F*F3F27&F0F-F4F5-%&STYLEG6#%&PATCHG-%(SCAL INGG6#%,CONSTRAINEDG-%+LIGHTMODELG6#%(LIGHT_3G-%+PROJECTIONG6%$\"#SF)$ \"#gF)\"\"\"" 1 2 0 1 10 0 2 1 5 1 1 1.000000 60.000000 40.000000 1 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Take the following wedge which has half the volume of the origi nal cuboid." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 268 "wedge:=[[C1,D1,A1,B1],[C1,D1,D2,C2],\n[A1,B1,C2,D2 ],[A1,D1,D2],[B1,C1,C2]]:\np1 := plots[polygonplot3d](Cuboid,style=wir eframe,color=black):\np2 := plots[polygonplot3d](wedge,style=patch,lig htmodel=light3):\nplots[display]([p1,p2],scaling=constrained,orientati on=[40,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6' -%)POLYGONSG6*7&7%$\"\"!F)F(F(7%$\"\"%F)F(F(7%F+$\"\"&F)F(7%F(F.F(7&7% F(F(F+7%F+F(F+7%F+F.F+7%F(F.F+7&F'F0F5F27&F*F-F4F37&F'F*F3F27&F0F-F4F5 -%'COLOURG6&%$RGBGF)F)F)-%&STYLEG6#%%LINEG-F$6(F8F&7&F3F2F0F-7%F3F*F-7 %F2F'F0-F?6#%&PATCHG-%(SCALINGG6#%,CONSTRAINEDG-%+LIGHTMODELG6#%(LIGHT _3G-%+PROJECTIONG6%$\"#SF)$\"#gF)\"\"\"" 1 2 0 1 10 0 2 1 5 1 1 1.000000 60.000000 40.000000 1 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Bisect the wedge into two pyramids, one with a rectangular base (on its side) and the other with a triangular base." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "triangle := [A1,C2,D1]:\np1 := plo ts[polygonplot3d](wedge,style=wireframe,color=black):\np2 := plots[pol ygonplot3d](triangle,style=patch,lightmodel=light3):\nplots[display]([ p1,p2],scaling=constrained,orientation=[60,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6)7&7%$\"\"!F)F(F(7%$ \"\"%F)F(F(7%F+F(F+7%F(F(F+7&F'F*7%F+$\"\"&F)F(7%F(F1F(7&F-F.F3F07%F-F *F07%F.F'F3-%'COLOURG6&%$RGBGF)F)F)-%&STYLEG6#%%LINEG-F$6$7%F-F3F*-F<6 #%&PATCHG-%(SCALINGG6#%,CONSTRAINEDG-%+LIGHTMODELG6#%(LIGHT_3G-%+PROJE CTIONG6%$\"#gF)FP\"\"\"" 1 2 0 1 10 0 2 1 5 1 1 1.000000 60.000000 60.000000 1 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "The animation shows the two pyramids movi ng apart." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "h := 0.1:\npyrmd1 := [[A1,B1,C1,D1],[B1,C2,C1],[B1,C 2,A1],[A1,C2,D1],[C1,C2,D1]]:\npyrmd2 := [[A1,C2,D1],[D1,C2,D2],[A1,C2 ,D2],[A1,D2,D1]]:\np1 := plots[polygonplot3d](pyrmd1):\np2 := plots[po lygonplot3d](pyrmd2):\nanim := [seq(plots[display]([p1,plottools[trans late](p2,h*k,h*k,-h*k)]),k=0..20)]:\nplots[display](anim,scaling=const rained,orientation=[60,60],\n style=patch,lightmodel=light3,insequen ce=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Finally bisect the pyramid with a square base into two pyramids with congruent triangula r bases. " }}{PARA 0 "" 0 "" {TEXT -1 40 "These two pyramids have the \+ same volume." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "pyrmd1 := [[A1,B1,C1,D1],[B1,C2,C1],[B1,C2,A1],[A 1,C2,D1],[C1,C2,D1]]:\ntriangle := [A1,C2,C1]:\np1 := plots[polygonplo t3d](pyrmd1,style=wireframe,thickness=1,color=black):\np2 := plots[pol ygonplot3d](triangle,style=patch,lightmodel=light3):\nplots[display]([ p1,p2],scaling=constrained,orientation=[70,50]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6*7&7%$\"\"%\"\"!$F*F *F(7%F+F+F(7%F+F+F+7%F(F+F+7%F,7%F+$\"\"&F*F+F-7%F,F0F'7%F'F0F.7%F-F0F .-%'COLOURG6&%$RGBGF*F*F*-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"\"-F$6$7 %F'F0F--F;6#%&PATCHG-%(SCALINGG6#%,CONSTRAINEDG-%+LIGHTMODELG6#%(LIGHT _3G-%+PROJECTIONG6%$\"#qF*$\"#]F*FA" 1 2 0 1 10 0 2 1 5 1 1 1.000000 50.000000 70.000000 1 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The animation shows these two p yramids moving apart." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "h := 0.1:\npyrmd1 := [[A1,B1,C1],[A1,C2, C1],[B1,C2,C1],[A1,C2,B1]]:\npyrmd2 := [[A1,C1,D1],[A1,C2,C1],[C1,C2,D 1],[A1,C2,D1]]:\np1 := plots[polygonplot3d](pyrmd1):\np2 := plots[poly gonplot3d](pyrmd2):\nanim := [seq(plots[display]([p1,plottools[transla te](p2,h*k,h*k,-h*k)]),k=0..20)]:\nplots[display](anim,scaling=constra ined,orientation=[100,60],\n style=patch,lightmodel=light3,insequenc e=true);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "The bottom triangular face of the lower pyramid is congruent to t he base of the first triangular pyramid cut off from the wedge." }} {PARA 0 "" 0 "" {TEXT -1 79 "Since these two pyramids also have the sa me height, they have the same volume. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 81 "The last animation shows the wedge ex ploding into three pyramids of equal volume." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 527 "h := 0.1:\npyrmd1 := [[A1,B1,C1],[A1,C2,C1],[B1,C2,C1],[A1,C2,B1]]:\npyrmd2 := [[A1,C1, D1],[A1,C2,C1],[C1,C2,D1],[A1,C2,D1]]:\npyrmd3 := [[A1,C2,D1],[D1,C2,D 2],[A1,C2,D2],[A1,D2,D1]]:\np1 := plots[polygonplot3d](pyrmd1):\np2 := plots[polygonplot3d](pyrmd2):\np3 := plots[polygonplot3d](pyrmd3):\na nim := [seq(plots[display]([plottools[translate](p1,-h*k,-h*k,h*k),p2, \n plottools[translate](p3,h*k,h*k,-h*k)]),k=0..20)]:\nplots[display ](anim,scaling=constrained,orientation=[100,60],\n style=patch,light model=light3,insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT3D 486 486 486 {PLOTDATA 3 "6'-%(ANIMATEG677%-%)POLYGONSG6&7%7%$\"\"%\"\"!$F.F.F, 7%F/F/F,7%F/F/F/7%F+7%F/$\"\"&F.F/F17%F0F3F17%F+F3F0-F(6&7%F+F17%F,F/F 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2" "Curve 3" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 81 "The wedge has now been divided into three pyramids which all have the same volume" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 51 "If the three dimensions of the original \+ cuboid are " }{TEXT 297 1 "a" }{TEXT -1 2 ", " }{TEXT 298 1 "b" } {TEXT -1 5 " and " }{TEXT 299 1 "c" }{TEXT -1 75 ", then the volume of each of the three pyramids obtained from the wedge is " }{XPPEDIT 18 0 "a*b*c/6;" "6#**%\"aG\"\"\"%\"bGF%%\"cGF%\"\"'!\"\"" }{TEXT -1 2 ". \n" }{TEXT 258 0 "" }{TEXT -1 11 " This is " }{XPPEDIT 18 0 "base*ar ea/3;" "6#*(%%baseG\"\"\"%%areaGF%\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 300 1 "x" }{TEXT -1 1 " " }{TEXT 263 2 "he" }{TEXT 272 0 "" }{TEXT 271 5 "ight." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 249 "The final three pyramids which have arisen in the dissection o f the wedge have some special characteristics with regard to the angle s between their faces and edges. One of them has three faces which are mutually perpendicular right-angled triangles." }}{PARA 0 "" 0 "" {TEXT -1 171 "However, as is illustrated in the previous section, a ge neral triangular pyramid can be sheered in two stages into a pyramid \+ which has three mutually perpendicular faces." }}{PARA 0 "" 0 "" {TEXT -1 13 "The formula " }{XPPEDIT 18 0 "volume = base*area/3;" "6# /%'volumeG*(%%baseG\"\"\"%%areaGF'\"\"$!\"\"" }{TEXT -1 2 " " }{TEXT 301 1 "x" }{TEXT 265 8 " height" }{TEXT -1 22 " therefore applies to \+ " }{TEXT 259 22 "any triangular pyramid" }{TEXT -1 4 " or " }{TEXT 259 11 "tetrahedron" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 272 "Since the polygonal base of a general p yramid can be be subdivided into triangles, the whole pyramid can also be subdivided into triangular pyramids all having the same upper vert ex as the original pyramid and bases given by the triangles of the sub divided polygonal base." }}{PARA 0 "" 0 "" {TEXT -1 46 "This enables u s to conclude that the formula " }{XPPEDIT 18 0 "volume = base*area/3 ;" "6#/%'volumeG*(%%baseG\"\"\"%%areaGF'\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 305 1 "x" }{TEXT -1 1 " " }{TEXT 264 6 "height" }{TEXT -1 12 " a pplies to " }{TEXT 259 11 "any pyramid" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The volume of a cone" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 185 "If we take pyramids which have bases whi ch are regular polygons with a progressively larger number of sides, t he resulting solids become progressively more like a cone. We say that the " }{TEXT 259 14 "limiting solid" }{TEXT -1 11 " is a cone." }} {PARA 0 "" 0 "" {TEXT -1 41 "For example, provided that the procedure \+ " }{TEXT 0 7 "pyramid" }{TEXT -1 5 " . . " }{HYPERLNK 17 "pyramid" 1 " " "pyramid" }{TEXT -1 100 " is active, you can draw the following pyra mid with a base which is a regular polygon with 20 sides." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plots [display](pyramid([0,0,0],20,1,3),scaling=constrained,style=patch,orie ntation=[25,65]);" }}{PARA 13 "" 1 "" {GLPLOT3D 487 487 487 {PLOTDATA 3 "6&-%)POLYGONSG67767%$\"+j^c5&*!#5$\"+W*p,4$F*$\"\"!F.7%$\"+V*p,4)F* $\"+CD&y(eF*F-7%$\"+AD&y(eF*$\"+W*p,4)F*F-7%$\"+Q*p,4$F*$\"+l^c5&*F*F- 7%F-$\"\"\"F.F-7%$!+Q*p,4$F*FF]o7%F>FAF]o7%FAFDF]o7% FDFGF]o7%FGFJF]o7%FJFMF]o7%FMFPF]o7%FPFSF]o7%FSFVF]o7%FVFYF]o7%FYFfnF] o7%FfnFgnF]o7%FgnFhnF]o7%FhnFinF]o7%FinFjnF]o7%FjnF[oF]o7%F[oF'F]o-%(S CALINGG6#%,CONSTRAINEDG-%&STYLEG6#%&PATCHG-%+PROJECTIONG6%$\"#DF.$\"#l F.F@" 1 2 0 1 10 0 2 1 1 1 1 1.000000 65.000000 25.000000 1 0 "Curve 1 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Since the formula " } {XPPEDIT 18 0 "volume = base*area/3;" "6#/%'volumeG*(%%baseG\"\"\"%%ar eaGF'\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 302 1 "x" }{TEXT -1 1 " " } {TEXT 266 6 "height" }{TEXT -1 67 " applies to all pyramids, it must a lso apply to the limiting solid." }}{PARA 0 "" 0 "" {TEXT -1 44 "There fore the volume of a cone is given by " }{XPPEDIT 18 0 "V = Pi*r^2*h/ 3;" "6#/%\"VG**%#PiG\"\"\"*$%\"rG\"\"#F'%\"hGF'\"\"$!\"\"" }{TEXT -1 8 ", where " }{TEXT 287 1 "r" }{TEXT -1 31 " is the radius of the base and " }{TEXT 288 1 "h" }{TEXT -1 15 " is the height." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "The surface area of a sphere" }} {PARA 0 "" 0 "" {TEXT -1 32 "The surface area of a sphere is " } {XPPEDIT 18 0 "S = 4*Pi*r^2;" "6#/%\"SG*(\"\"%\"\"\"%#PiGF'%\"rG\"\"# " }{TEXT -1 8 ", where " }{TEXT 289 1 "r" }{TEXT -1 29 " is the radius of the sphere." }}{PARA 0 "" 0 "" {TEXT -1 60 "It is the same as the \+ surface area of a cylnder with radius " }{TEXT 290 1 "r" }{TEXT -1 56 " and with axis passing through the centre of the sphere." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "p1 : = plots[display](plottools[sphere](),style=patchnogrid,lightmodel=ligh t4):\np2 := plots[tubeplot]([0,0,t],t=-1..1,radius=1,tubepoints=40,num points=30,\n style=wireframe,color=gray):\nplots[display]([p1, p2],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 435 366 366 {PLOTDATA 3 "6&-%)POLYGONSG6]_mX,6\"6\"F'[gl'!%\"!!#-\"%\"$00000000000 0000000000000000000003FF00000000000003FC0B5150F6DA2CC00000000000000003 FEFB9EA92EC689C3FC023581EF7D7F63FA14BF61DCEDA9F3FEFB9EA92EC689C0000000 00000000000000000000000003FF0000000000000X,F'F(F'[gl'!%\"!!#-\"%\"$3FC 0B5150F6DA2CC00000000000000003FEFB9EA92EC689C3FD0907DC193068C000000000 00000003FEEE8DD4748BF153FCFFFFFFFFFFFF83FB126145E9ECD4D3FEEE8DD4748BF1 53FC023581EF7D7F63FA14BF61DCEDA9F3FEFB9EA92EC689CX,F'F(F'[gl'!%\"!!#- 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FhjmF`\\n7%FcamF[[nF`\\n7%F^[nF`[nF`\\n7%Fc[nFe[nF`\\n7%Fc`mFh[nF`\\n7 %F\\`mF[\\nF`\\n7J7%F\\`mF_`m$!3E8C " 0 "" {MPLTEXT 1 0 685 "specs1:=radius=sqrt(1-t^2), tubepoints=40,lightmodel=light4:\nspecs2:=radius=1,tubepoints=40:\np1 \+ := plots[tubeplot]([0,0,t],t=-1..0.5,specs1,numpoints=30,style=wirefra me):\np2 := plots[tubeplot]([0,0,t],t=0.5..0.8,specs1,numpoints=6,styl e=patchnogrid):\np3 := plots[tubeplot]([0,0,t],t=0.8..1,specs1,numpoin ts=4,style=wireframe):\np4 := plots[tubeplot]([0,0,t],t=-1..0.5,specs2 ,numpoints=30,\n style=wireframe,color=gray):\np5 := plots[t ubeplot]([0,0,t],t=0.5..0.8,specs2,numpoints=6,\n style=patc hnogrid,color=navy):\np6 := plots[tubeplot]([0,0,t],t=0.8..1,specs2,nu mpoints=4,\n style=wireframe,color=gray):\nplots[display]([p 1,p2,p3,p4,p5,p6],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 445 455 455 {PLOTDATA 3 "6*-%%MESHG6$7@7J7%$\"\"!F*F)$!\"\"F*F(F(F(F(F (F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(7 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"Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The \+ coloured rings on the sphere and the cylinder each have area " } {XPPEDIT 18 0 "2*Pi*r*h;" "6#**\"\"#\"\"\"%#PiGF%%\"rGF%%\"hGF%" } {TEXT -1 8 ", where " }{TEXT 292 1 "r" }{TEXT -1 46 " is the radius of the sphere and cylinder and " }{TEXT 293 1 "h" }{TEXT -1 43 " is the \+ height of the ring on the cylinder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 48 "We start by trying to see why this is tr ue when " }{TEXT 291 1 "h" }{TEXT -1 10 " is small." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 310 253 253 {PLOTDATA 2 " 69-%%TEXTG6$7$$\"$N#!\"#$\"#V!\"\"%\"AG-F$6$7$F'$\"$:&F)%\"BG-F$6$7$$ \"$l$F)F1%\"CG-F$6$7$F7$\"$N%F)%\"DG-F$6$7$F7$\"$s$F)%\"EG-F$6$7$$\"#J F,$\"$:%F)%\"PG-F$6$7$$\"\"$\"\"!$!#8F)%\"QG-F$6$7$$!#:F)FT%\"OG-F$6$7 $$\"\"&FSFT%\"XG-%'CURVESG6%7$7$FSFjn7$$\"1+++++++DFenFjn-%'COLOURG6&% 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}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 246 "We can think of a sphere as being th e limit of certain polyhedra as the number of faces becomes progressiv ely larger. Indeed the following animated sequence of tubeplots approx imating a sphere actually consist of polyhedra with rectangular faces. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "anim := NULL:\nfor k from 8 to 32 by 4 do \n h := Pi/k:\n \+ c := seq(evalf(cos(h*i)),i=0..2*k):\n s := seq(evalf(sin(h*i)),i=0 ..2*k):\n m := [seq([seq([s[j]*c[i],s[j]*s[i],c[j]],i=1..2*k+1)],j=1 ..k+1)]:\n anim := anim,PLOT3D(MESH(m)):\nend do:\nplots[display]([a nim],lightmodel=light1,scaling=constrained,\n style=patch, insequence=true,orientation=[45,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "Pick one of these approximating polyh edra . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "h := Pi/20:\nc := seq(evalf(cos(h*k)),k=0..40):\ns : = seq(evalf(sin(h*k)),k=0..40):\nm := [seq([seq([s[j]*c[i],s[j]*s[i],c [j]],i=1..41)],j=1..21)]:\np := PLOT3D(MESH(m)):\nplots[display](p,lig htmodel=light1,scaling=constrained,\n style= patch,orientation=[45,65]);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 93 ". . and construct pyramids from each rect angular face with apex at the centre of the sphere. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "C:=[0,0,0 ]:\nP:=m[4,1]: Q:=m[4,2]: \nR:=m[5,2]: S:=m[5,1]:\npyrmd := plots[poly gonplot3d]([[P,Q,R,S],[P,C,Q],[Q,C,R],[R,C,S],[S,C,P]]):\np := plots[d isplay](p,style=wireframe):\nplots[display]([p,pyrmd],lightmodel=light 1,scaling=constrained,\n orientation=[-65,50] );" }}{PARA 13 "" 1 "" {GLPLOT3D 499 449 449 {PLOTDATA 3 "6'-%%MESHG6$ 777K7%$\"\"!F*F)$\"\"\"F*F(F(F(F(F(F(F(F(F(F(7%$!\"!F*F)F+F-F-F-F-F-F- F-F-F-7%F.F.F+F0F0F0F0F0F0F0F07%F)F.F+F1F1F1F1F1F1F1F1F1F(7K7%$\"+^YMk :!#5F)$\"+1M)o()*F67%$\"+s\\3X:F6$\"+(=urW#!#6F77%$\"+u,y([\"F6$\"+B34 M[F>F77%$\"+!HTQR\"F6$\"++h(>5(F>F77%$\"+39el7F6$\"+br)\\>*F>F77%$\"+6 (eh5\"F6FOF77%$\"+_r)\\>*F>FJF77%$\"+&4w>5(F>FEF77%$\"+834M[F>F@F77%$ \"+*=urW#F>F:F77%F)F4F77%$!+*=urW#F>F:F77%$!+834M[F>F@F77%$!+&4w>5(F>F EF77%$!+_r)\\>*F>FJF77%$!+6(eh5\"F6FOF77%$!+39el7F6FLF77%$!+!HTQR\"F6F GF77%$!+u,y([\"F6FBF77%$!+s\\3X:F6F F77%F^p$!+B34M[F>F77%F[p$!++h(>5(F>F77%Fho$!+br)\\>*F>F77%FeoFeoF77%Fb oFhoF77%F_oF[pF77%F\\oF^pF77%FinFapF77%F)FdpF77%FenFapF77%FXF^pF77%FUF [pF77%FRFhoF77%FOFeoF77%FJF`qF77%FEF]qF77%F@FjpF77%F:FgpF7F37K7%$\"+W* p,4$F6F)$\"+j^c5&*F67%$\"+C[7_IF6FBFer7%$\"+ii#*QHF6$\"+$G]\"\\&*F>Fer 7%$\"+\"ehLv#F6$\"+(z2HS\"F6Fer7%$\"+++++DF6$\"+?jN;=F6Fer7%$\"+A,3&=# F6FjsFer7%FgsFesFer7%$\"+'z2HS\"F6F`sFer7%$\"+k-:\\&*F>F[sFer7%$\"+E34 M[F>FhrFer7%F)FcrFer7%$!+E34M[F>FhrFer7%$!+k-:\\&*F>F[sFer7%$!+'z2HS\" F6F`sFer7%$!+?jN;=F6FesFer7%$!+A,3&=#F6FjsFer7%$!+++++DF6FgsFer7%$!+\" ehLv#F6FbsFer7%$!+ii#*QHF6F]sFer7%$!+C[7_IF6FBFer7%$!+W*p,4$F6F)Fer7%F `vFjpFer7%F]v$!+$G]\"\\&*F>Fer7%Fju$!+(z2HS\"F6Fer7%FguFauFer7%FduFduF er7%FauFguFer7%F^uFjuFer7%F[uF]vFer7%FhtF`vFer7%F)FcvFer7%FdtF`vFer7%F atF]vFer7%F^tFjuFer7%FgsFguFer7%FjsFduFer7%FesFauFer7%F`sFjvFer7%F[sFg vFer7%FhrFjpFerFbr7K7%$\"+)*\\!*RXF6F)$\"+U_15*)F67%$\"+M7,%[%F6FGF`x7 %$\"+Kiq5KF6FcyF`x7%$\"+/#*[oEF6F^yF`x7%$\"+QP2h?F6$\"+t\\3XSF6F `x7%$\"+&z2HS\"F6$\"+Liq5(F>FcxF`x7%F)F^xF`x7%$!+/h(> 5(F>FcxF`x7%$!+&z2HS\"F6F`zF`x7%$!+QP2h?F6F[zF`x7%$!+/#*[oEF6F^yF`x7%$ !+4w>5KF6FcyF`x7%$!+'HgGn$F6F`yF`x7%$!+s\\3XSF6F[yF`x7%$!+Kiq*F>Ff^lFc^l7% F)Fa^lFc^l7%$!+hr)\\>*F>Ff^lFc^l7%$!+5KF6FidlF`dl7%$\"+=,3&=#F6 $\"+>^)\\s'F6F`dl7%FOFcdlF`dl7%F)F`dlF`dl7%FeoFcdlF`dl7%$!+=,3&=#F6F\\ flF`dl7%$!+2w>5KF6FidlF`dl7%$!+w$pi:%F6FdelF`dl7%$!+(*******\\F6F_elF` dl7%$!+ESh?dF6Fd_lF`dl7%$!+_vO+jF6FcyF`dl7%$!+=^)\\s'F6FjsF`dl7%$!+K7, %)pF6FOF`dl7%$!+5y1rqF6F)F`dl7%FgglFeoF`dl7%FdglFduF`dl7%FaglFc[lF`dl7 %F^glFdalF`dl7%F[glF[glF`dl7%Fhfl$!+FSh?dF6F`dl7%FeflFaglF`dl7%Fbfl$!+ >^)\\s'F6F`dl7%FeoFgglF`dl7%F)FjglF`dl7%FOFgglF`dl7%FjelFfhlF`dl7%Fgel FaglF`dl7%FbelFbhlF`dl7%F_elF[glF`dl7%F\\elFdalF`dl7%FidlFc[lF`dl7%Ffd lFduF`dl7%FcdlFeoF`dlF_dl7K7%$\"+W*p,4)F6F)$\"+AD&y(eF67%$\"+Flc!*zF6F JFgil7%$\"+V)3Up(F6FesFgil7%$\"+-UR3sF6F^yFgil7%$\"+r\\3XlF6F__lFgil7% FdelFdelFgil7%$\"+\"e#GbZF6$\"+s\\3XlF6Fgil7%$\"+%HgGn$F6$\"+.UR3sF6Fg il7%$\"+&******\\#F6$\"+X)3Up(F6Fgil7%$\"+49el7F6FjilFgil7%F)FeilFgil7 %$!+49el7F6FjilFgil7%$!+&******\\#F6Fc[mFgil7%$!+%HgGn$F6F^[mFgil7%$!+ \"e#GbZF6FijlFgil7%FbhlFdelFgil7%$!+r\\3XlF6F__lFgil7%$!+-UR3sF6F^yFgi l7%$!+V)3Up(F6FesFgil7%$!+Flc!*zF6FJFgil7%$!+W*p,4)F6F)Fgil7%F`]mFhoFg il7%F]]mFguFgil7%Fj\\mFf[lFgil7%Fg\\mFgalFgil7%FbhlFbhlFgil7%Fc\\m$!+s \\3XlF6Fgil7%F`\\m$!+.UR3sF6Fgil7%F]\\m$!+X)3Up(F6Fgil7%Fj[mF`]mFgil7% F)Fc]mFgil7%Ff[mF`]mFgil7%Fa[mFa^mFgil7%F\\[mF^^mFgil7%FgjlF[^mFgil7%F delFbhlFgil7%FcjlFgalFgil7%F`jlFf[lFgil7%F]jlFguFgil7%FjilFhoFgilFdil7 K7%$\"+V_15*)F6F)$\"+&*\\!*RXF67%$\"+avO+))F6FEFb_m7%$\"+5c(RZ)F6F`sFb _m7%$\"+ji#*QzF6F[zFb_m7%F`jlF\\`lFb_m7%FidlFidlFb_m7%$\"+Y\\?P_F6F^[m Fb_m7%$\"+q\\3XSF6F[`mFb_m7%$\"+w:O`FF6$\"+7c(RZ)F6Fb_m7%$\"+\"HTQR\"F 6Fe_mFb_m7%F)F`_mFb_m7%$!+\"HTQR\"F6Fe_mFb_m7%$!+w:O`FF6Fh`mFb_m7%$!+q \\3XSF6F[`mFb_m7%$!+Y\\?P_F6F^[mFb_m7%FaglFidlFb_m7%Fj\\mF\\`lFb_m7%$! +ji#*QzF6F[zFb_m7%$!+5c(RZ)F6F`sFb_m7%$!+avO+))F6FEFb_m7%$!+V_15*)F6F) Fb_m7%FcbmF[pFb_m7%F`bmFjuFb_m7%F]bmF_]lFb_m7%Fj\\mF^clFb_m7%FaglFaglF b_m7%FhamF^^mFb_m7%FeamF]bmFb_m7%Fbam$!+7c(RZ)F6Fb_m7%F_amFcbmFb_m7%F) FfbmFb_m7%F[amFcbmFb_m7%Ff`mF`cmFb_m7%Fc`mF]bmFb_m7%F``mF^^mFb_m7%Fidl FaglFb_m7%F`jlF^clFb_m7%F[`mF_]lFb_m7%Fh_mFjuFb_m7%Fe_mF[pFb_mF__m7K7% $\"+l^c5&*F6F)$\"+Q*p,4$F67%$\"+EVZ$R*F6F@Fadm7%$\"+u\\3X!*F6F[sFadm7% $\"+6c(RZ)F6F`zFadm7%$\"+W)3Up(F6Fa`lFadm7%F\\flF\\flFadm7%$\"+W*p,f&F 6Fc[mFadm7%$\"+Iiq " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The volume of each of these pyramids wi th apex at the centre of the sphere is" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 267 24 "area of rectangul ar base" }{TEXT -1 2 " " }{TEXT 303 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "radius*of*sphere/3;" "6#**%'radiusG\"\"\"%#ofGF%%'sphereGF%\"\"$ !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 96 "Adding up the v olumes of these pyramids gives an approximate value for the volume of \+ a sphere as" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{TEXT 268 22 "surfac e area of sphere" }{TEXT -1 1 " " }{TEXT 304 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "radius*of*sphere/3;" "6#**%'radiusG\"\"\"%#ofGF%%'spher eGF%\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 177 "Taking progressively smaller rectangles gives progressively better approxima tions, so the formula just given must give the exact volume of a spher e in terms of the surface area." }}{PARA 0 "" 0 "" {TEXT -1 47 "Since \+ the surface area of a sphere with radius " }{TEXT 296 1 "r" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "4*Pi*r^2;" "6#*(\"\"%\"\"\"%#PiGF%%\"rG\"\"# " }{TEXT -1 18 ", the volume is " }{XPPEDIT 18 0 "4*Pi*r^3/3;" "6#** \"\"%\"\"\"%#PiGF%%\"rG\"\"$F(!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Find the volum e of a pyramid which has a square base with each side of length 1.738 \+ cm. and a height of 4.528 cm." }}{PARA 0 "" 0 "" {TEXT -1 32 "_______ _________________________" }}{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 32 "________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 " Find t he volume and surface area of a sphere with radius 3.609 m." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 32 "____________________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "Find the volume and surface area of a cyl inder with radius 0.4958 cm. and height 6.947 cm." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 32 "________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Find the volume of a cone with base radius 2.0325 cm. \+ and height 8.273 cm." }}{PARA 0 "" 0 "" {TEXT -1 32 "_________________ _______________" }}{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 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 211 "A tank i s in the shape of a cylinder with hemispherical ends. The length of th e tank including the hemispherical ends is 4.529 m and the radius of t he cylindrical part is 0.8372 m. 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