document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=901768>Question 901768</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During a hurricane, a telegraph pole was broken in such a way that the top struck the level ground at a distance of 20 feet from the base of the pole. It was replaced by an identical pole which was broken by another gale at a point 5 feet lower and the top struck the ground a distance of 30 feet from the base. What was the original height of the poles? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #546976 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=546976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i grapahed the equations and then i solved them algebraically for you.<BR>\n" );
document.write( "they both point to the same solution which is a good sign.\r<BR>\n" );
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document.write( "the solution is:\r<BR>\n" );
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document.write( "the base of the pole from the original break was 21 feet and the piece of the pole that fell to the ground was 29 feet for a total of 50 feet in length.\r<BR>\n" );
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document.write( "the base of the pole from the replacement break was 16 feet and the piece of the pole that fell to the ground was 34 feet for a total of 50 feet again.\r<BR>\n" );
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document.write( "the equations were set up as follows:\r<BR>\n" );
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document.write( "two right triangles were formed.\r<BR>\n" );
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document.write( "The first case triangle had the following characteristics.<BR>\n" );
document.write( "x = vertical leg of the triangle = bottom of the pole that is still connected to the base.<BR>\n" );
document.write( "y = hypotenuse of the triangle = top of the pole that fell over.<BR>\n" );
document.write( "20 = horizontal leg of the triangle = the distance from the top of the pole that is touching the ground to the base of the pole.\r<BR>\n" );
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document.write( "the second case triangle had the following characteristics.<BR>\n" );
document.write( "x-5 = vertical leg of the triangle = bottom of the pole that is still connected to the base.<BR>\n" );
document.write( "y+5 = hypotenuse of the triangle = top of the pole that fell over.<BR>\n" );
document.write( "30 = horizontal leg of the triangle = the distance from the top of the pole that is touching the ground to the base of the pole.\r<BR>\n" );
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document.write( "the pythagorean formula was used on both triangles.\r<BR>\n" );
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document.write( "first case triangle is x^2 + 20^2 = y^2<BR>\n" );
document.write( "second case triangle is (x-5)^2 + 30^2 = (y+5)^2\r<BR>\n" );
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document.write( "these 2 equations were graphed and the intersection of the two graphs is the solution.  \r<BR>\n" );
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document.write( "that solution is that x = 21 and y = 29 as shown on the graph.\r<BR>\n" );
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document.write( "the graphical solution is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/091504.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the algebraic solution was solved as follows:\r<BR>\n" );
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document.write( "both equations needed to be solved simultaneously since the same solution for x and y had to be common to both equations.\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "x^2 + 20^2 = y^2<BR>\n" );
document.write( "(x-5)^2 + 30^2 = (y+5)^2\r<BR>\n" );
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document.write( "the vertical leg of the replacement pole was 5 feet shorter than the vertical leg of the original pole which is the reason for (x-5) as the vertical leg.\r<BR>\n" );
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document.write( "since the vertical leg was 5 feet shorter, the upper part that fell had to be 5 feet longer which is the reason for (y+5) as the hypotenuse.\r<BR>\n" );
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document.write( "simplify these equations to get:\r<BR>\n" );
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document.write( "x^2 + 400 = y^2<BR>\n" );
document.write( "(x-5)^2 + 900 = (y+5)^2\r<BR>\n" );
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document.write( "solve for y^2 and y from the first equation to get:\r<BR>\n" );
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document.write( "y^2 = x^2 + 400<BR>\n" );
document.write( "y = sqrt(x^2 + 400)\r<BR>\n" );
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document.write( "simplify the second case equation of (x-5)^2 + 900 = (y+5)^2 to get:\r<BR>\n" );
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document.write( "x^2 - 10x + 25 + 900 = y^2 + 10y + 25\r<BR>\n" );
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document.write( "replace y^2 with x^2 + 400 and replace y with sqrt(x^2 + 400) to get:\r<BR>\n" );
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document.write( "x^2 - 10x + 25 + 900 = x^2 + 400 + 10 * sqrt(x^2 + 400) + 25\r<BR>\n" );
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document.write( "subtract x^2 and subtract 25 from both sides of the equation and subtract 400 from both sides of the equation to get:\r<BR>\n" );
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document.write( "-10x + 500 = 10*sqrt(x^2 + 400)\r<BR>\n" );
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document.write( "divide both sides of the equation by 10 to get:\r<BR>\n" );
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document.write( "-x + 50 = sqrt(x^2 + 400)\r<BR>\n" );
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document.write( "rearrange the terms on the left side of the equation to get:\r<BR>\n" );
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document.write( "50 - x = sqrt(x^2 + 400)\r<BR>\n" );
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document.write( "square both sides of the equation to get:\r<BR>\n" );
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document.write( "2500 - 100x + x^2 = x^2 + 400\r<BR>\n" );
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document.write( "subtract x^2 from both sides of the equation and subtract 400 from both sides of the equation and add 100x to both sides of the equation to get:\r<BR>\n" );
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document.write( "2100 = 100x\r<BR>\n" );
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document.write( "divide both sides of the equation by 100 to get:\r<BR>\n" );
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document.write( "21 = x\r<BR>\n" );
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document.write( "that's the length of the vertical leg of the first case triangle.\r<BR>\n" );
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document.write( "using pythagorus again, you get:<BR>\n" );
document.write( "21^2 + 20^2 = y^2<BR>\n" );
document.write( "solve for y^2 to get:<BR>\n" );
document.write( "y^2 = 841<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = 29\r<BR>\n" );
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document.write( "the original pole was broken 21 feet above the ground and the top of the pole that fell over was 29 feet in length for a total length of 50 feet.\r<BR>\n" );
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document.write( "the replacement pole was broken 5 feet further down, so the replacement pole was broken 16 feet above the ground the the top of the pole was 34 feet in length for a total length of 50 feet again.\r<BR>\n" );
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document.write( "since x = 21 and y = 29, then (x-5) = 16 and (y+5) = 34.<BR>\n" );
document.write( "21^2 + 20^2 = 29^2<BR>\n" );
document.write( "16^2 + 30^2 = 34^2<BR>\n" );
document.write( "solving these equations gets:<BR>\n" );
document.write( "841 = 841<BR>\n" );
document.write( "1156 = 1156<BR>\n" );
document.write( "this confirms the solution is correct with x = 21 and y = 29\r<BR>\n" );
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document.write( "the requested solution to the problem is:\r<BR>\n" );
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document.write( "the original height of the poles was 50 feet.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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