document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=194310>Question 194310</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 #145806 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=145806\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1 \r<BR>\n" );
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document.write( "part i)\r<BR>\n" );
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document.write( "If you look at angles HBD, BGI, and GCA, you'll see that these angles form a triangle. Remember, the sum of all the angles in a triangle is ALWAYS 180 degrees.\r<BR>\n" );
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document.write( "So < HBD + < BGI + < GCA = 180\r<BR>\n" );
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document.write( "part ii)\r<BR>\n" );
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document.write( "Similar to part i), the segments BG, GC, and CB form a triangle also. The union of these three segments will simply result in triangle BGC\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \bar{BG} \cup \bar{GC} \cup \bar{CB} = \bigtriangleup BGC\">\r<BR>\n" );
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document.write( "iii)\r<BR>\n" );
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document.write( "The angle GCB and the segment BH only have the points B and G in common\r<BR>\n" );
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document.write( "So the answer is simply the two points B and G.\r<BR>\n" );
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document.write( "iv)\r<BR>\n" );
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document.write( "The angle ABG and the segment BD only have the point B in common\r<BR>\n" );
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document.write( "So the answer is simply the point B.\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "a) \r<BR>\n" );
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document.write( "Supplementary angles are angles that add up to 180 degrees. In a visual sense, if you place the angles right next to one another (where the vertices meet and one sides touch), then the two angles will form a straight line.\r<BR>\n" );
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document.write( "So algebraically, this means that if \"x\" and \"y\" are two unknown angles, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=180 ALIGN=MIDDLE  ALT=\"x%2By=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "However, we do know that one angle is 34 degrees. So let's plug this into \"y\" (it doesn't matter which variable we choose)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B34=180 ALIGN=MIDDLE  ALT=\"x%2B34=180\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now subtract 34 from both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=180-34 ALIGN=MIDDLE  ALT=\"x=180-34\"   DISABLED_style= \"max-width:90%; height:auto;\" > and combine like terms to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=146 ALIGN=MIDDLE  ALT=\"x=146\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the supplementary angle to 34 degrees is 146 degrees.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Let x and y be the angle measurements for angles ABC and PQR respectively\r<BR>\n" );
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document.write( "Since \"< ABC and < PQR are supplementary angles\", this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=180 ALIGN=MIDDLE  ALT=\"x%2By=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > (see above). Also, because \"< ABC is eleven times as large as < PQR\", we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11y ALIGN=MIDDLE  ALT=\"x=11y\"   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%2By=180 ALIGN=MIDDLE  ALT=\"x%2By=180\"   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=11y%2By=180 ALIGN=MIDDLE  ALT=\"11y%2By=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11y ALIGN=MIDDLE  ALT=\"x=11y\"   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=12y=180 ALIGN=MIDDLE  ALT=\"12y=180\"   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=%28180%29%2F%2812%29 ALIGN=MIDDLE  ALT=\"y=%28180%29%2F%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <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 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=15 ALIGN=MIDDLE  ALT=\"y=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce. So the first angle ABC is 15 degrees.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11y ALIGN=MIDDLE  ALT=\"x=11y\"   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=11%2815%29 ALIGN=MIDDLE  ALT=\"x=11%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=15 ALIGN=MIDDLE  ALT=\"y=15\"   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=165 ALIGN=MIDDLE  ALT=\"x=165\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply. So the second angle PQR is 165 degrees.\r<BR>\n" );
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document.write( "# 3 \r<BR>\n" );
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document.write( "Looking at angle 3, we can see that is a vertical angle to the angle labeled with 29 degrees. Since vertical angles are equal, this means that \r<BR>\n" );
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document.write( "angle 3 = 29 degrees\r<BR>\n" );
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document.write( "Now recall that if two parallel lines are cut with a transversal lines, then alternate interior angles are equal to each other. Because angles 3 and 4 are alternate interior angles, this means that\r<BR>\n" );
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document.write( "angle 3 = angle 4 = 29 degrees\r<BR>\n" );
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document.write( "Since angles 4 and 7 are vertical angles, this means that\r<BR>\n" );
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document.write( "angle 4 = angle 7 = 29 degrees\r<BR>\n" );
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document.write( "So the following angles are 29 degrees: angles 3, 4, and 7\r<BR>\n" );
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document.write( "Now since the top angle labeled 29 degrees and angle 1 form the line \"l\", this means that \r<BR>\n" );
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document.write( "29 + angle 1 = 180 degrees\r<BR>\n" );
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document.write( "Solving for angle 1, we get\r<BR>\n" );
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document.write( "angle 1 = 151\r<BR>\n" );
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document.write( "Angles 1 and 2 are vertical angles. So using the same logic, this means \r<BR>\n" );
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document.write( "angle 1 = angle 2 = 151 degrees\r<BR>\n" );
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document.write( "Angles 2 and 5 are another pair of alternate interior angles, so \r<BR>\n" );
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document.write( "angle 2 = angle 5 = 151 degrees\r<BR>\n" );
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document.write( "Finally, because angles 5 and 6 are vertical angles, this tells us that\r<BR>\n" );
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document.write( "angle 5 = angle 6 = 151 degrees\r<BR>\n" );
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document.write( "So the angles 1, 2, 5, and 6 all have measures of 151 degrees.\r<BR>\n" );
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document.write( "# 4\r<BR>\n" );
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document.write( "First, we need to draw a picture to see what's going on:\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%2C7%29%2C%0D%0Alocate%281%2C2%2C25%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: the longest side of ANY triangle is the hypotenuse.\r<BR>\n" );
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document.write( "So to find the area and perimeter, we need to find the length of the unknown side.\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=7 ALIGN=MIDDLE  ALT=\"7\"   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=7 ALIGN=MIDDLE  ALT=\"b=7\"   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=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=25 ALIGN=MIDDLE  ALT=\"c=25\"   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%2B7%5E2=25%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B7%5E2=25%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=7 ALIGN=MIDDLE  ALT=\"b=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=25 ALIGN=MIDDLE  ALT=\"c=25\"   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=576 ALIGN=MIDDLE  ALT=\"x%5E2=576\"   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%28576%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28576%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=24 ALIGN=MIDDLE  ALT=\"x=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root.\r<BR>\n" );
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document.write( "So the unknown side is 24 cm long.\r<BR>\n" );
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document.write( "Now let's find the perimeter. The perimeter of any triangle is \r<BR>\n" );
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document.write( "Perimeter = Side 1 + Side 2 + Side 3\r<BR>\n" );
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document.write( "So \r<BR>\n" );
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document.write( "Perimeter = 7 + 24 + 25\r<BR>\n" );
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document.write( "Perimeter = 56\r<BR>\n" );
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document.write( "which means that the perimeter is 56 cm\r<BR>\n" );
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document.write( "Now onto the area. The area of any triangle is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%2824%2A7%29%2F2 ALIGN=MIDDLE  ALT=\"A=%2824%2A7%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=7 ALIGN=MIDDLE  ALT=\"h=7\"   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=168%2F2 ALIGN=MIDDLE  ALT=\"A=168%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\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=84 ALIGN=MIDDLE  ALT=\"A=84\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the area of the triangle is 84 square cm.\r<BR>\n" );
document.write( "<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( "<BR>\n" );
document.write( "\n" );
document.write( "First, recall (or look up) that there are 12 inches in a foot. So we have the equation \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "12 inches = 1 foot\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So to convert from inches to feet, simply divide by 12. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42%2F12=3.5 ALIGN=MIDDLE  ALT=\"42%2F12=3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "42 inches = 3.5 feet\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So there are 42 inches in 3 and a half feet.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the diameter is 3.5 feet, this means that half of this figure is the radius which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.5%2F2=1.75 ALIGN=MIDDLE  ALT=\"3.5%2F2=1.75\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1.75 ALIGN=MIDDLE  ALT=\"r=1.75\"   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( "<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 area of a circle 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%2A%281.75%29%5E2 ALIGN=MIDDLE  ALT=\"A=pi%2A%281.75%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1.75 ALIGN=MIDDLE  ALT=\"r=1.75\"   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=3.0625pi ALIGN=MIDDLE  ALT=\"A=3.0625pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 1.75 to get 3.0625 and 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=3.0625pi ALIGN=MIDDLE  ALT=\"3.0625pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > square feet.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the answer is C)\r<BR>\n" );
document.write( "<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( "\n" );
document.write( "The volume of a rectangular prism/box is: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Volume = Area of base * Height\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this means that the formula is \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=Ah ALIGN=MIDDLE  ALT=\"V=Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > where \"A\" is the area of the base.\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=Ah ALIGN=MIDDLE  ALT=\"V=Ah\"   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=1.2%2A2 ALIGN=MIDDLE  ALT=\"V=1.2%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1.2 ALIGN=MIDDLE  ALT=\"A=1.2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   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=2.4 ALIGN=MIDDLE  ALT=\"V=2.4\"   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 volume is 2.4 cubic meters which makes the answer choice B)\r<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( "Since the diameter is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=6 ALIGN=MIDDLE  ALT=\"d=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=d%2F2=6%2F2=3 ALIGN=MIDDLE  ALT=\"r=d%2F2=6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or simply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3 ALIGN=MIDDLE  ALT=\"r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Recall, the formula for the volume of a cone is \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=%281%2F3%29%2Api%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2Api%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=%281%2F3%29%2Api%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2Api%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=%281%2F3%29%2Api%2A3%5E2%2A10 ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2Api%2A3%5E2%2A10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3 ALIGN=MIDDLE  ALT=\"r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=10 ALIGN=MIDDLE  ALT=\"h=10\"   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=%281%2F3%29%2Api%2A9%2A10 ALIGN=MIDDLE  ALT=\"V=%281%2F3%29%2Api%2A9%2A10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 3 to get 9.\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=%2890%2F3%29%2Api ALIGN=MIDDLE  ALT=\"V=%2890%2F3%29%2Api\"   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=30pi ALIGN=MIDDLE  ALT=\"V=30pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\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=94.248 ALIGN=MIDDLE  ALT=\"V=94.248\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use 3.14 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the volume of the tank is about 94.247 cubic feet. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since 1 cubic foot = 7.480519 gallons\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means that we simply need to multiply 94.247 by 7.480519 to get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "94.247*7.480519=705.0164\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now round to the nearest hundredth to get 705.02\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the tank can hold about 705.02 gallons.\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( "\n" );
document.write( "The volume of a pyramid with a square or rectangular base is \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V=%28L%2AW%2AH%29%2F3 ALIGN=MIDDLE  ALT=\"V=%28L%2AW%2AH%29%2F3\"   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=%28L%2AW%2AH%29%2F3 ALIGN=MIDDLE  ALT=\"V=%28L%2AW%2AH%29%2F3\"   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=%2811%2A11%2A12%29%2F3 ALIGN=MIDDLE  ALT=\"V=%2811%2A11%2A12%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=11 ALIGN=MIDDLE  ALT=\"L=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=11 ALIGN=MIDDLE  ALT=\"W=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=12 ALIGN=MIDDLE  ALT=\"H=12\"   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=%281452%29%2F3 ALIGN=MIDDLE  ALT=\"V=%281452%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\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=484 ALIGN=MIDDLE  ALT=\"V=484\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the volume is 484 cubic meters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=======================================================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "# 9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Because the diameter is 12 feet, this means that the radius is 6 feet (since the radius is half the diameter).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Remember, the formula for the volume of a cylinder is:\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;\" >\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%2A6%5E2%2A10 ALIGN=MIDDLE  ALT=\"V=pi%2A6%5E2%2A10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=6 ALIGN=MIDDLE  ALT=\"r=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=10 ALIGN=MIDDLE  ALT=\"h=10\"   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%2A36%2A10 ALIGN=MIDDLE  ALT=\"V=pi%2A36%2A10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36\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=360pi ALIGN=MIDDLE  ALT=\"V=360pi\"   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( "So the <i>exact</i> volume is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360pi ALIGN=MIDDLE  ALT=\"360pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we 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 3.14159, we 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=360%283.14159%29=1130.9724 ALIGN=MIDDLE  ALT=\"360%283.14159%29=1130.9724\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now round to the nearest hundredth to get 1130.97\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the <i>approximate</i> volume is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1130.97 ALIGN=MIDDLE  ALT=\"1130.97\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=======================================================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "# 10\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Area of a trapezoid:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28h%28b%5B1%5D%2Bb%5B2%5D%29%29%2F2 ALIGN=MIDDLE  ALT=\"A=%28h%28b%5B1%5D%2Bb%5B2%5D%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D ALIGN=MIDDLE  ALT=\"b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D ALIGN=MIDDLE  ALT=\"b%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the parallel bases.\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=%28h%28b%5B1%5D%2Bb%5B2%5D%29%29%2F2 ALIGN=MIDDLE  ALT=\"A=%28h%28b%5B1%5D%2Bb%5B2%5D%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%286%2814%2B21%29%29%2F2 ALIGN=MIDDLE  ALT=\"A=%286%2814%2B21%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=6 ALIGN=MIDDLE  ALT=\"h=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D=14 ALIGN=MIDDLE  ALT=\"b%5B1%5D=14\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D=21 ALIGN=MIDDLE  ALT=\"b%5B2%5D=21\"   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=%286%2835%29%29%2F2 ALIGN=MIDDLE  ALT=\"A=%286%2835%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=210%2F2 ALIGN=MIDDLE  ALT=\"A=210%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=105 ALIGN=MIDDLE  ALT=\"A=105\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the area of the trapezoid is 115 square inches.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "One of the many properties of a parallelogram is that opposite parallel sides are equal. So the side parallel to the side length 9 inches is also 9 inches. Similarly, the side opposite the length 5 inches is also 5 inches.\r<BR>\n" );
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document.write( "So we have the sides are of 9, 5, 9, and 5 inches\r<BR>\n" );
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document.write( "Add these lengths up to get 9+5+9+5=14+14=28\r<BR>\n" );
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document.write( "So the perimeter of the parallelogram is 28 inches.\r<BR>\n" );
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document.write( "=======================================================================================================\r<BR>\n" );
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document.write( "# 11\r<BR>\n" );
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document.write( "First, let's find the area of the upper rectangle. Simply multiply the length 11 yd by the width 5 yd to get: 11*5=55. So the area of the upper rectangle is 55 square yards.\r<BR>\n" );
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document.write( "Now let's find the area of the triangle. Looking at the figure, we see that the base is 4 yards and the height is 5 yards. Now multiply these two dimensions and divide by 2 to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2A5%29%2F2=20%2F2=10 ALIGN=MIDDLE  ALT=\"%284%2A5%29%2F2=20%2F2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the area of the triangle is 10 square yards.\r<BR>\n" );
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document.write( "Finally, simply square the base of the lower square to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2=4%2A4=16 ALIGN=MIDDLE  ALT=\"4%5E2=4%2A4=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the total area is: 55+10+16=81 square yards which means that 81 square yards of carpeting will be needed. \r<BR>\n" );
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document.write( "Since \"Carpeting will cost $8.00 a square yard\", this means that we simply multiply the total area by the cost per square yard. So...\r<BR>\n" );
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document.write( "81*8=648\r<BR>\n" );
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document.write( "which means that the total cost will be $648\r<BR>\n" );
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document.write( "# 12\r<BR>\n" );
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document.write( "Because the two triangles are congruent, this means that their corresponding parts are equal. So their two longer legs are equal which means that AB = DE = 24 cm\r<BR>\n" );
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document.write( "This means that AB = 24 cm\r<BR>\n" );
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document.write( "Also, their two hypotenuses are equal. So AC = DF = 25 cm which means that DF = 25 cm\r<BR>\n" );
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document.write( "Also, congruent triangles have equal corresponding angles. Since angle BAC is the angle between the longer leg and the hypotenuse for the first triangle and angle EFD is the same for the second triangle, this makes \r<BR>\n" );
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document.write( "angle BAC = angle EFD = 16 degrees\r<BR>\n" );
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document.write( "So angle BAC = 16 degrees\r<BR>\n" );
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document.write( "The same applies to angles ACB and EFD. These angles are also corresponding angles. So this means that\r<BR>\n" );
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document.write( "angle ACB = angle EFD = 74 degrees\r<BR>\n" );
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document.write( "which means that angle EFD = 74 degrees\r<BR>\n" );
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document.write( "Now from before, it was stated that the sum of all angles in a triangle add to 180 degrees. Since we know two angles of the first triangle, this means that we can find the remaining angle by first adding the two angles to get: 74+16=90\r<BR>\n" );
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document.write( "Now subtract this result from 180 to get 180 - 90 = 90\r<BR>\n" );
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document.write( "So the remaining angle ABC is 90 degrees which means angle ABC = 90\r<BR>\n" );
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document.write( "Because angles ABC and DEF are corresponding angles, this tells us that angle ABC = angle DEF = 90\r<BR>\n" );
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document.write( "So angle DEF = 90\r<BR>\n" );
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document.write( "# 13\r<BR>\n" );
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document.write( "Start with five randomly drawn points. These are the vertices\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/vertices.png\">\r<BR>\n" );
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document.write( "Now simply connect every vertex with a line. Make sure that EVERY point has two lines connecting to it (so a bridge doesn't form). In addition to connecting every vertex, draw a line from any one vertex to itself to form a loop (in green):\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/vertices2.png\">\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );