document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1050269>Question 1050269</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A goat is tied outside the fence of a 5-m diameter circular garden using a 3-m rope. One end of the rope is tied to the collar around the neck of the goat and the other s attached to the fence at the point which is at the level as the goat's collar (the goat can move anywhere within 3 meters from the outside of the circular fence). Find the area by which the goat can move around by integration. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #666290 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=666290\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=fc9.JPG\">.<BR>\n" );
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document.write( "The red circle is the circular garden.<BR>\n" );
document.write( "The purple circle is the 3m radius from the edge of the circular garden fence.<BR>\n" );
document.write( "The equations of the two circles are,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=%282.5%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=%282.5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=6.25 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=6.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2.5%29%5E2%2By%5E2=%283%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-2.5%29%5E2%2By%5E2=%283%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2.5%29%5E2%2By%5E2=9 ALIGN=MIDDLE  ALT=\"%28x-2.5%29%5E2%2By%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find the intersection points substitute from eq. 1 into eq. 2,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=6.25-x%5E2 ALIGN=MIDDLE  ALT=\"y%5E2=6.25-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2.5%29%5E2%2B6.25-x%5E2=9 ALIGN=MIDDLE  ALT=\"%28x-2.5%29%5E2%2B6.25-x%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5x%2B6.25%2B6.25-x%5E2=9 ALIGN=MIDDLE  ALT=\"x%5E2-5x%2B6.25%2B6.25-x%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5x%2B12.5-x%5E2=9 ALIGN=MIDDLE  ALT=\"x%5E2-5x%2B12.5-x%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5x=-3.5 ALIGN=MIDDLE  ALT=\"-5x=-3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.7 ALIGN=MIDDLE  ALT=\"x=0.7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then integrate to get the sliver area (shown in green),<BR>\n" );
document.write( "the limits of integration would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.7 ALIGN=MIDDLE  ALT=\"x=0.7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.5 ALIGN=MIDDLE  ALT=\"x=2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( ".<BR>\n" );
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document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=int%28%28y%5B2%5D-y%5B1%5D%29%2Cdx%2Cx=0.7%2C2.5%29 ALIGN=MIDDLE  ALT=\"A=int%28%28y%5B2%5D-y%5B1%5D%29%2Cdx%2Cx=0.7%2C2.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=int%28%28sqrt%289-%28x-2.5%29%5E2%29-sqrt%286.25-x%5E2%29%29%2Cdx%2Cx=0.7%2C2.5%29 ALIGN=MIDDLE  ALT=\"A=int%28%28sqrt%289-%28x-2.5%29%5E2%29-sqrt%286.25-x%5E2%29%29%2Cdx%2Cx=0.7%2C2.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1.874 ALIGN=MIDDLE  ALT=\"A=1.874\"   DISABLED_style= \"max-width:90%; height:auto;\" > <-- Obtained numerically<BR>\n" );
document.write( "Double that plus half of the area of the large circle will be the complete area that the goat can graze,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Btot%5D=2%281.874%29%2B%28pi%283%29%5E2%29%2F2 ALIGN=MIDDLE  ALT=\"A%5Btot%5D=2%281.874%29%2B%28pi%283%29%5E2%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Btot%5D=17.89 ALIGN=MIDDLE  ALT=\"A%5Btot%5D=17.89\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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