document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=193878>Question 193878</A>: <I><SPAN STYLE=\"word-break: break-all;\">  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #145558 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=145558\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "Let \r<BR>\n" );
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document.write( "x = measure of < PQR and \r<BR>\n" );
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document.write( "y = measure of < ABC\r<BR>\n" );
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document.write( "Since these two angles are complementary angles, this means that the two angles add to 90 degrees. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=90 ALIGN=MIDDLE  ALT=\"x%2By=90\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's label this equation 1.\r<BR>\n" );
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document.write( "Also, because \"< PQR is nine times as large as < ABC\", we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9y ALIGN=MIDDLE  ALT=\"x=9y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We'll call this equation 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=90 ALIGN=MIDDLE  ALT=\"x%2By=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9y%2By=90 ALIGN=MIDDLE  ALT=\"9y%2By=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9y ALIGN=MIDDLE  ALT=\"x=9y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10y=90 ALIGN=MIDDLE  ALT=\"10y=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the left side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2890%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"y=%2890%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=9 ALIGN=MIDDLE  ALT=\"y=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9y ALIGN=MIDDLE  ALT=\"x=9y\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the second equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9%289%29 ALIGN=MIDDLE  ALT=\"x=9%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=9 ALIGN=MIDDLE  ALT=\"y=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=81 ALIGN=MIDDLE  ALT=\"x=81\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=81 ALIGN=MIDDLE  ALT=\"x=81\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=9 ALIGN=MIDDLE  ALT=\"y=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that the measures of < PQR and < ABC are 81 and 9 degrees respectively.\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "Take note that the angle marked 68 degrees and angle 3 are vertical angles. So this means that these two angles are equal.\r<BR>\n" );
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document.write( "So angle 3 is also 68 degrees.\r<BR>\n" );
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document.write( "Now because lines l and m are parallel, this means that the alternate interior angles are equal. Since angles 3 and 5 are alternate interior angles, this tells us that\r<BR>\n" );
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document.write( "angle 3 = angle 5\r<BR>\n" );
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document.write( "So angle 5 is 68 degrees as well.\r<BR>\n" );
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document.write( "Now remember, the sum of the 3 angles of ANY triangle is ALWAYS 180 degrees. So if we let the measure of angle 6 be \"x\", this means that\r<BR>\n" );
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document.write( "40 + angle 5 + angle 6 = 180\r<BR>\n" );
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document.write( "Plug in the given values to get:\r<BR>\n" );
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document.write( "40 + 68 + x = 180\r<BR>\n" );
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document.write( "Combine like terms to get \r<BR>\n" );
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document.write( "108 + x = 180\r<BR>\n" );
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document.write( "Subtract 108 from both sides \r<BR>\n" );
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document.write( "x = 180 - 108\r<BR>\n" );
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document.write( "and combine like terms to get\r<BR>\n" );
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document.write( "x = 72\r<BR>\n" );
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document.write( "So the measure of angle 6 is 72 degrees.\r<BR>\n" );
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document.write( "# 3\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "First, let \"x\" be the length of AC to get the triangle:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-0.5%2C2%2C-0.5%2C3.2%2C%0D%0Aline%280%2C0%2C0%2C3%29%2C%0D%0Aline%280%2C3%2C2%2C0%29%2C%0D%0Aline%282%2C0%2C0%2C0%29%2C%0D%0Alocate%28-0.2%2C1.5%2Cx%29%2C%0D%0Alocate%281%2C-0.2%2C12%29%2C%0D%0Alocate%281%2C2%2C13%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To find the unknown length, we need to use the Pythagorean Theorem.\r<BR>\n" );
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document.write( "Remember, the Pythagorean Theorem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where \"a\" and \"b\" are the legs of a triangle and \"c\" is the hypotenuse.\r<BR>\n" );
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document.write( "Since the legs are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=x ALIGN=MIDDLE  ALT=\"a=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=12 ALIGN=MIDDLE  ALT=\"b=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "   \r<BR>\n" );
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document.write( "Also, since the hypotenuse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=13 ALIGN=MIDDLE  ALT=\"c=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the Pythagorean theorem.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B12%5E2=13%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B12%5E2=13%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=x ALIGN=MIDDLE  ALT=\"a=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=12 ALIGN=MIDDLE  ALT=\"b=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=13 ALIGN=MIDDLE  ALT=\"c=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B144=13%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B144=13%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144 ALIGN=MIDDLE  ALT=\"144\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B144=169 ALIGN=MIDDLE  ALT=\"x%5E2%2B144=169\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=169 ALIGN=MIDDLE  ALT=\"169\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=169-144 ALIGN=MIDDLE  ALT=\"x%5E2=169-144\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144 ALIGN=MIDDLE  ALT=\"144\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=25 ALIGN=MIDDLE  ALT=\"x%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%2825%29 ALIGN=MIDDLE  ALT=\"x=sqrt%2825%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 25 to get 5.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that AC = 5.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Now that we know the three sides are of length 5, 12, and 13 units, simply add them up to get\r<BR>\n" );
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document.write( "Perimeter = 5 + 12 + 13 = 17 + 13 = 30\r<BR>\n" );
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document.write( "So the perimeter is 30 units.\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "Recall, the area of a triangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28bh%29%2F2 ALIGN=MIDDLE  ALT=\"A=%28bh%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28bh%29%2F2 ALIGN=MIDDLE  ALT=\"A=%28bh%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%2812%2A5%29%2F2 ALIGN=MIDDLE  ALT=\"A=%2812%2A5%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given base and height.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=60%2F2 ALIGN=MIDDLE  ALT=\"A=60%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=30 ALIGN=MIDDLE  ALT=\"A=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide.\r<BR>\n" );
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document.write( "So the area is 30 square units.\r<BR>\n" );
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document.write( "Note: this is just a coincidence that the perimeter and the area are equal.\r<BR>\n" );
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document.write( "# 4\r<BR>\n" );
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document.write( "First, we need to find the length of diagonal.\r<BR>\n" );
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document.write( "If we cut the garden in half along the diagonal, we'll have this triangle set up (where \"x\" is the length of the diagonal):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-0.5%2C2%2C-0.5%2C3.2%2C%0D%0Aline%280%2C0%2C0%2C3%29%2C%0D%0Aline%280%2C3%2C2%2C0%29%2C%0D%0Aline%282%2C0%2C0%2C0%29%2C%0D%0Alocate%28-0.2%2C1.5%2C18%29%2C%0D%0Alocate%281%2C-0.2%2C24%29%2C%0D%0Alocate%281%2C2%2Cx%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Remember, the Pythagorean Theorem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where \"a\" and \"b\" are the legs of a triangle and \"c\" is the hypotenuse.\r<BR>\n" );
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document.write( "Since the legs are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24 ALIGN=MIDDLE  ALT=\"24\"   DISABLED_style= \"max-width:90%; height:auto;\" > this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=18 ALIGN=MIDDLE  ALT=\"a=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=24 ALIGN=MIDDLE  ALT=\"b=24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "   \r<BR>\n" );
document.write( "\n" );
document.write( "Also, since the hypotenuse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=x ALIGN=MIDDLE  ALT=\"c=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the Pythagorean theorem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%5E2%2B24%5E2=x%5E2 ALIGN=MIDDLE  ALT=\"18%5E2%2B24%5E2=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=18 ALIGN=MIDDLE  ALT=\"a=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=24 ALIGN=MIDDLE  ALT=\"b=24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=x ALIGN=MIDDLE  ALT=\"c=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=324%2B24%5E2=x%5E2 ALIGN=MIDDLE  ALT=\"324%2B24%5E2=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=324 ALIGN=MIDDLE  ALT=\"324\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=324%2B576=x%5E2 ALIGN=MIDDLE  ALT=\"324%2B576=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24 ALIGN=MIDDLE  ALT=\"24\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900=x%5E2 ALIGN=MIDDLE  ALT=\"900=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=900 ALIGN=MIDDLE  ALT=\"x%5E2=900\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28900%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28900%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=30 ALIGN=MIDDLE  ALT=\"x=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 900 to get 30.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the diagonal is 30 meters long. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now just multiply the length of the diagonal by the cost per linear meter to get: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2A12=360 ALIGN=MIDDLE  ALT=\"30%2A12=360\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So it will cost $360 to build the diagonal walkway.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "# 5\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "First, let's convert every given measurement to feet. Recall (or look up), the given equality: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1 yard = 3 feet\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words, there are 3 feet in a yard\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So to convert 16 yards to feet, simply multiply both sides of the equation given above by 16 to get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1*16 yards = 3*16 feet\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "16 yards = 48 feet\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So there are 48 feet in 16 yards\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now go back to the original equation, but now multiply both sides by 70 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1*70 yards = 3 * 70 feet\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "70 yards = 210 feet\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So there are 210 feet in 70 yards\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: to convert yards to feet, all we're doing is multiplying the given measurement in yards by 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the equivalent dimensions (in feet) are 48 feet by 210 feet\r<BR>\n" );
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document.write( "\n" );
document.write( "------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's find the area of the lawn:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The formula for the area of a rectangle is:\r<BR>\n" );
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document.write( "\n" );
document.write( " Area = Length * Width \r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which looks like: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=LW ALIGN=MIDDLE  ALT=\"A=LW\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=LW ALIGN=MIDDLE  ALT=\"A=LW\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=48%2A210 ALIGN=MIDDLE  ALT=\"A=48%2A210\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given dimensions (in feet)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10800 ALIGN=MIDDLE  ALT=\"A=10800\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the area of the lawn is 10,800 square feet.\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now divide the area of the lawn by the area that ONE bag can cover to get:\r<BR>\n" );
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document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10800%2F525=20.571 ALIGN=MIDDLE  ALT=\"10800%2F525=20.571\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So it will take 21 bags to cover his lawn (I rounded to the nearest whole number since you can't buy half a bag)\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now multiply this figure by the cost per bag to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2A5.75=120.75 ALIGN=MIDDLE  ALT=\"21%2A5.75=120.75\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So it will cost him $120.75 to cover his lawn with fertilizer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "# 6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: since the diameter is 22mm, we take half of this number to find the radius. So the radius is 11 mm (since 11 is half of 22)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Area:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Recall, the area of a circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=pi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"A=pi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=pi%2A11%5E2 ALIGN=MIDDLE  ALT=\"A=pi%2A11%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=11 ALIGN=MIDDLE  ALT=\"r=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > (see above)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=121pi ALIGN=MIDDLE  ALT=\"A=121pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 11 to get 121. Also, rearrange the terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the exact area is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=121pi ALIGN=MIDDLE  ALT=\"121pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > square milimeters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is irrational, we cannot write out the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we must leave it alone.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Circumference:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Remember, the circumference of a circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=2%2Ar%2Api ALIGN=MIDDLE  ALT=\"C=2%2Ar%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=2%2Ar%2Api ALIGN=MIDDLE  ALT=\"C=2%2Ar%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=2%2A11%2Api ALIGN=MIDDLE  ALT=\"C=2%2A11%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=11 ALIGN=MIDDLE  ALT=\"r=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > (see above)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=22%2Api ALIGN=MIDDLE  ALT=\"C=22%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 11 to get 22\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the exact circumference is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22pi ALIGN=MIDDLE  ALT=\"22pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > milimeters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "# 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: since 10 mm = 1 cm, this means that 5 cm = 50 mm (just multiply both sides by 5)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the diameter is 50 mm, this means that the radius is 25 mm (the radius is half the diameter)\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The formula for the volume of a cylinder is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"V=pi%2Ar%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"V=pi%2Ar%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2A25%5E2%2A140 ALIGN=MIDDLE  ALT=\"V=pi%2A25%5E2%2A140\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=25 ALIGN=MIDDLE  ALT=\"r=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=140 ALIGN=MIDDLE  ALT=\"h=140\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=pi%2A625%2A140 ALIGN=MIDDLE  ALT=\"V=pi%2A625%2A140\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 25 to get 625\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=87500pi ALIGN=MIDDLE  ALT=\"V=87500pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply and rearrange the terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=274889.356 ALIGN=MIDDLE  ALT=\"V=274889.356\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > with 3.14159265... and multiply. Note: this result is approximate\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the glass can hold about 274,889.356 cubic milimeters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now divide the result by 1000 to convert to convert to cubic centimeters to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=274889.356%2F1000=274.889 ALIGN=MIDDLE  ALT=\"274889.356%2F1000=274.889\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now round to the nearest tenth to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=274.889 ALIGN=MIDDLE  ALT=\"V=274.889\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the glass can hold about 274.9 cubic centimeters of water.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==============================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "# 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's some terminology:\r<BR>\n" );
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document.write( "\n" );
document.write( "Vertex: A point representing a location (city, town, etc...). In general, what the vertex represents isn't important.\r<BR>\n" );
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document.write( "Edge: A line connecting two vertices (plural for vertex)\r<BR>\n" );
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document.write( "Odd Vertex: A vertex with an odd number of edges connecting to it\r<BR>\n" );
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document.write( "Even Vertex: A vertex with an even number of edges connecting to it\r<BR>\n" );
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document.write( "So here's one way to draw the graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/graph22.png\">\r<BR>\n" );
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document.write( "Take note that...\r<BR>\n" );
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document.write( "Vertex A: even vertex with 2 edges connecting to it\r<BR>\n" );
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document.write( "Vertex B: odd vertex with 3 edges connecting to it\r<BR>\n" );
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document.write( "Vertex C: even vertex with 2 edges connecting to it\r<BR>\n" );
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document.write( "Vertex D: odd vertex with 3 edges connecting to it\r<BR>\n" );
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document.write( "\n" );
document.write( "==============================================================================\r<BR>\n" );
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document.write( "# 9\r<BR>\n" );
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document.write( "It is possible to travel through every door once (and only once) given that you either start in room F or end in room F (otherwise you can't do it).\r<BR>\n" );
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document.write( "So here are 6 possible scenarios (there are possibly more)\r<BR>\n" );
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document.write( "Scenario # 1:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/9solution1.png\">\r<BR>\n" );
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document.write( "--------------------------------\r<BR>\n" );
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document.write( "Scenario # 2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/9solution2.png\">\r<BR>\n" );
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document.write( "--------------------------------\r<BR>\n" );
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document.write( "Scenario # 3:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/9solution3.png\">\r<BR>\n" );
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document.write( "--------------------------------\r<BR>\n" );
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document.write( "Scenario # 4:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/9solution4.png\">\r<BR>\n" );
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document.write( "\n" );
document.write( "--------------------------------\r<BR>\n" );
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document.write( "\n" );
document.write( "Scenario # 5:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/9solution5.png\">\r<BR>\n" );
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document.write( "--------------------------------\r<BR>\n" );
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document.write( "Scenario # 6:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/work/9solution6.png\">\r<BR>\n" );
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document.write( "--------------------------------\r<BR>\n" );
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document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );