document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1029851>Question 1029851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Because of prevailing winds, a tree grew so that it was leaning 4 degrees from vertical. At a point 35 meters from the tree the angle of elevation to the top of the tree is 23 degrees. Find the height of the tree. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #644901 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=644901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the diagram i worked with is shown here.<BR>\n" );
document.write( "look below the diagram for the discussion that follows.<BR>\n" );
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document.write( "the right triangle formed is ACD.<BR>\n" );
document.write( "the triangle formed that goes to the height of the tree is ABD.<BR>\n" );
document.write( "drop a perpendicular from B to E on the line AD.<BR>\n" );
document.write( "you have two right triangles formed.<BR>\n" );
document.write( "they are ABE and DBE.<BR>\n" );
document.write( "angle BAE is 23 degrees.<BR>\n" );
document.write( "angle ABE is 67 degrees.<BR>\n" );
document.write( "angle AEB is 90 degrees.<BR>\n" );
document.write( "angle BDC is 4 degrees because this is the angle from the vertical that the tree is slanted at.<BR>\n" );
document.write( "since angle CDE is a right angle, then angle BDE is 86 degrees.<BR>\n" );
document.write( "since triangle DBE is a right triangle, than angle DBE is 4 degrees.<BR>\n" );
document.write( "line BE is the altitude of both triangles ABE and DBE.<BR>\n" );
document.write( "the line AD is broken up into two parts.<BR>\n" );
document.write( "the first part is line AE whose length is designated as x.<BR>\n" );
document.write( "the second part is line ED whose length is designated as 35-x.<BR>\n" );
document.write( "AE and ED together form the line AD which is given as 35 meters long.<BR>\n" );
document.write( "the length of the line BE is designated as h.\r<BR>\n" );
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document.write( "from this we derive the following:\r<BR>\n" );
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document.write( "tan(23) = h/x\r<BR>\n" );
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document.write( "tan(86) = h/(35-x)\r<BR>\n" );
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document.write( "solve for h in both these formulas and you get:\r<BR>\n" );
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document.write( "h = x * tan(23)<BR>\n" );
document.write( "h = (35-x) * tan(86)\r<BR>\n" );
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document.write( "this gets you x * tan(23) = (35-x) * tan(86)<BR>\n" );
document.write( "simplify this equation to get x * tan(23) = 35 * tan(86) - x * tan(86)<BR>\n" );
document.write( "add x * tan(86) to both sides of this equation to get x * tan(23) + x * tan(86) = 35 * tan(86).<BR>\n" );
document.write( "factor out the x to get x * (tan(23) + tan(86)) = 35 * tan(86).<BR>\n" );
document.write( "divide both sides of this equation by (tan(23) + tan(86)) to get:<BR>\n" );
document.write( "x = (35 * tan(86) / (tan(23) + tan(86)).<BR>\n" );
document.write( "this results in x = 33.99107122.<BR>\n" );
document.write( "this makes 35 - x = 1.008928777.\r<BR>\n" );
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document.write( "h is calculated to be 33.99107122 * tan(23) and h is also calculated to be 1.008928777 * tan(86).\r<BR>\n" );
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document.write( "both formulas tell you that h is equal to 14.42835371 meters.\r<BR>\n" );
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document.write( "that's the height of the tree from the ground which is the length of line BE in the diagram.\r<BR>\n" );
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document.write( "i believe the actual height of the tree would be the length of the line BD.\r<BR>\n" );
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document.write( "that measures the length from the base of the tree to the top of the tree.\r<BR>\n" );
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document.write( "that can be calculated based on the following formulas.\r<BR>\n" );
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document.write( "sin(86) = BE / BD.<BR>\n" );
document.write( "solve for BD to get BD = BE / sin(86).\r<BR>\n" );
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document.write( "cos(86) = ED / BD.<BR>\n" );
document.write( "solve for BD to get BD = ED / cos(86).\r<BR>\n" );
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document.write( "BE^2 + ED^2 = BD^2 which makes BD = sqrt(BE^2 + ED^2).\r<BR>\n" );
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document.write( "all of these point to the same length of the line BD which is equal to 14.46358628 meters.\r<BR>\n" );
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document.write( "the vertical height of the tree is equal to 14.42835371 meters which is the length of the line BE.\r<BR>\n" );
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document.write( "what i think is the actual height of the tree is equal to 14.46358628 meters which is the length of the line BD.\r<BR>\n" );
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document.write( "that's what i get.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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