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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 23 "Dr. Sarah's Maple Demo \+
" }}{PARA 257 "" 0 "" {TEXT -1 2 "A " }{TEXT 256 0 "" }{TEXT -1 68 "Ci
rcle and Rectangle Can Never Have Both the Same Area and Perimeter" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 261 "" 0 "" {TEXT -1 69 "A Circle and a Square Can Never
 Have Both the Same Area and Perimeter" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 107 "To see that a circle and a square can never have both th
e same area and perimeter, we simultaneously solve " }}{PARA 0 "" 0 "
" {TEXT -1 110 "these conditions for the radius of the circle, r, and \+
the side length of the square, x.  Execute the following" }}{PARA 0 "
" 0 "" {TEXT -1 14 "Maple command:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "solve(\{x^2=Pi*r^2,4*x=2*Pi*r\},\{x,r\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$<$/%\"xG\"\"!/%\"rGF&F#" }}}{EXCHG {PARA 
259 "" 0 "" {TEXT -1 1 " " }{TEXT 259 41 "Notice that Maple outputs on
e solution.  " }}{PARA 260 "" 0 "" {TEXT -1 132 "Why doesn't the solut
ion give us a circle and square with equal area and perimeter.  What d
oes the solution look like geometrically?" }{TEXT 258 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Now we wi
ll investigate why a circle and rectangle can never both have the same
 area and perimeter.  " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Algebr
aic Explorations for Circles and Rectangles" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 194 "Let r be the radius of a circle, and x and y be the si
des of a rectangle.  If the area of the circle equals the area of the \+
rectangle, then we can solve for the condition on r that this imposes:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(\{x*y=Pi*r^2\},\{
r\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/%\"rG,$*&*$-%%sqrtG6#*(%#P
iG\"\"\"%\"xGF.%\"yGF.\"\"\"F1F-!\"\"!\"\"<#/F%F'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 129 "If the perimeter of the circle equals the perimet
er of the rectangle, then we can solve for the condition on r that thi
s imposes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(\{2*x+2
*y=2*Pi*r\},\{r\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/%\"rG*&,&%\"
xG\"\"\"%\"yGF)\"\"\"%#PiG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
67 "Since the radius of a circle can never be negative, using algebra,
 " }{TEXT 270 33 "we see that x+y must be equal to " }{XPPEDIT 271 0 "
sqrt(Pi*x*y);" "6#-%%sqrtG6#*(%#PiG\"\"\"%\"xGF(%\"yGF(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 44 "Can you solve for x and y
 by hand?  Explain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "solve(x+y=sqrt(x*y*Pi),\{x,y\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"yGF%/%\"xG*&,(%#PiG#\"\"\"\"\"#!
\"\"F,*&%\"IGF,-%%sqrtG6#,&*$)F*F-\"\"\"F.F*\"\"%F7#F.F-F,F%F,<$F$/F'*
&,(F*F+F.F,F/F+F,F%F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
265 38 "Notice that Maple uses I to represent " }{TEXT -1 0 "" }
{XPPEDIT 18 0 "sqrt(-1)" "6#-%%sqrtG6#,$\"\"\"!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 266 54 "If y is non-zero, why can't  x ever be \+
real?  Explain." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 267 168 "If y is 0, then x can be real.  Wha
t is x and r in this case?  Why doesn't this real solution yield a cir
cle and a rectangle with both have the same area and perimeter." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "
Graphical Explorations for Circle and Rectangle" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 76 "As above, if the perimeters and areas match up then \+
we know that x+y equals " }{XPPEDIT 18 0 "sqrt(Pi*x*y);" "6#-%%sqrtG6#
*(%#PiG\"\"\"%\"xGF(%\"yGF(" }}{PARA 0 "" 0 "" {TEXT -1 57 "We'll now \+
look at a 3d graph of the surfaces z=x+y and z=" }{XPPEDIT 18 0 "sqrt(
Pi*x*y)" "6#-%%sqrtG6#*(%#PiG\"\"\"%\"xGF(%\"yGF(" }}{PARA 0 "" 0 "" 
{TEXT -1 205 "with x values ranging from 1 to 4 and y values ranging f
rom 1 to 4.  Note that r=z, and so if we find an intersection point to
 the two surfaces, then that tells us that we have a solution for the \+
radius r." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot3d(\{x+y,s
qrt(x*y*Pi)\}, x=1..4, y=1..4);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%%GRI
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