document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=415189>Question 415189</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pole casts a shadow of 25ft at one time and a shadow of 10 ft at a later time when the angle of elevation is twice as large. Find the height of the pole, to the nearest foot. \r<BR>\n" );
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document.write( "I have tried using the double angle identity of tan and am stuck at 50h/625-h squared = h/10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #291182 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=291182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here's how i solved it.\r<BR>\n" );
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document.write( "x = angle of elevation.<BR>\n" );
document.write( "y = height of pole.\r<BR>\n" );
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document.write( "height of pole is equal to y\r<BR>\n" );
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document.write( "tan(x) = y/25\r<BR>\n" );
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document.write( "tan(2x) = y/10\r<BR>\n" );
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document.write( "the double angle tan formula is:\r<BR>\n" );
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document.write( "tan(2x) = 2tan(x) / (1 &#8722; tan^2(x))\r<BR>\n" );
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document.write( "we know that tan(2x) = y/10<BR>\n" );
document.write( "we know that tan(x) = y/25\r<BR>\n" );
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document.write( "substituting in tan(2x) = 2tan(x) / (1 &#8722; tan^2(x)), we get:\r<BR>\n" );
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document.write( "y/10 = (2y/25) / (1 - (y/25)^2)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "y/10 = (2y/25) / (1 - (y^2/625))\r<BR>\n" );
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document.write( "multiply both sides of the equation by (1 - (y^2/625) and divide both sides of the equation by (y/10) to get:\r<BR>\n" );
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document.write( "1 - (y^2/625) = (2y/25) / (y/10)\r<BR>\n" );
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document.write( "this is equivalent to:\r<BR>\n" );
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document.write( "1 - (y^2/625) = (2y/25) * (10/y)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "1 - (y^2/625) = 20/25\r<BR>\n" );
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document.write( "add (y^2/625) to both sides of the equation and subtract 20/25 from both sides of the equation to get:\r<BR>\n" );
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document.write( "1 - (20/25) = y^2/625\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "5/25 = y^2/625\r<BR>\n" );
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document.write( "multiply both sides of the equation by 625 to get:\r<BR>\n" );
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document.write( "(625*5)/25 = y^2\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "125 = y^2\r<BR>\n" );
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document.write( "take square root of both sides of the equation to get:\r<BR>\n" );
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document.write( "y = +/- 11.18033989\r<BR>\n" );
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document.write( "you have tan(x) = 11.18033989/25 = .447213596\r<BR>\n" );
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document.write( "arctan (.447213596) = 24.09484255 degrees = x.\r<BR>\n" );
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document.write( "you also have tan(2x) = 11.18033989/10 = 1.118033989\r<BR>\n" );
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document.write( "arctan(1.118033989) = 48.1896851 degrees = 2x\r<BR>\n" );
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document.write( "the results check out ok.\r<BR>\n" );
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document.write( "y = 11.18033989 feet which is the height of the pole.\r<BR>\n" );
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document.write( "formulas we started with are:\r<BR>\n" );
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document.write( "y = 25 * tan(x) and y = 10 * tan(2x)\r<BR>\n" );
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document.write( "our angles became:\r<BR>\n" );
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document.write( "x = 24.09484255 degrees<BR>\n" );
document.write( "2x = 48.1896851 degrees\r<BR>\n" );
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document.write( "tan(x) = .447213596<BR>\n" );
document.write( "tan(2x) = 1.118033989\r<BR>\n" );
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document.write( "y = 25*tan(x) = 25 * .447213596 = 11.18033989<BR>\n" );
document.write( "y = 10*tan(2x) = 10 * 1.118033989 = 11.18033989\r<BR>\n" );
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document.write( "height of the pole is 11.18033989 feet.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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