document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179789>Question 1179789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 300 ft from the base of a mountain the top and the base of a tower on its summit is sighted @ 27degree 43’ & 24degree 25’ respectively. At the base of the mountain the angles of elevation are 39degree & 35degree respectively. Find the height of the tower? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850174 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850174\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step using trigonometry.\r<BR>\n" );
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document.write( "**1. Diagram:**\r<BR>\n" );
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document.write( "* Draw a horizontal line representing the ground.<BR>\n" );
document.write( "* Mark a point A on the ground, 300 ft from the base of the mountain.<BR>\n" );
document.write( "* Draw a vertical line representing the mountain. Let the base of the mountain be B and the top be C.<BR>\n" );
document.write( "* Draw a tower on top of the mountain. Let the top of the tower be D.<BR>\n" );
document.write( "* From point A, draw lines to B, C, and D.<BR>\n" );
document.write( "* Draw a vertical line from the base of the mountain (B) to the top of the mountain (C). Also draw a vertical line from the base of the mountain to the top of the tower (D).\r<BR>\n" );
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document.write( "**2. Given Information:**\r<BR>\n" );
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document.write( "* Distance from A to the base of the mountain (AB) = 300 ft<BR>\n" );
document.write( "* Angle of elevation from A to the base of the tower (∠BAB) = 24°25'<BR>\n" );
document.write( "* Angle of elevation from A to the top of the tower (∠BAD) = 27°43'<BR>\n" );
document.write( "* Angle of elevation from B to the base of the tower (∠EBC) = 35°<BR>\n" );
document.write( "* Angle of elevation from B to the top of the tower (∠EBD) = 39° where E is a point on the horizontal line drawn from A to the base of the mountain.\r<BR>\n" );
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document.write( "**3. Convert Angles to Decimal Degrees:**\r<BR>\n" );
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document.write( "* 24°25' = 24 + 25/60 ≈ 24.4167°<BR>\n" );
document.write( "* 27°43' = 27 + 43/60 ≈ 27.7167°<BR>\n" );
document.write( "* 35°<BR>\n" );
document.write( "* 39°\r<BR>\n" );
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document.write( "**4. Calculate the Height of the Mountain (BC):**\r<BR>\n" );
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document.write( "* In triangle ABC, tan(24.4167°) = BC / AB<BR>\n" );
document.write( "* BC = AB * tan(24.4167°)<BR>\n" );
document.write( "* BC = 300 * tan(24.4167°)<BR>\n" );
document.write( "* BC ≈ 300 * 0.4539<BR>\n" );
document.write( "* BC ≈ 136.17 ft\r<BR>\n" );
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document.write( "**5. Calculate the Height of the Mountain Plus Tower (BD):**\r<BR>\n" );
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document.write( "* In triangle ABD, tan(27.7167°) = BD / AB<BR>\n" );
document.write( "* BD = AB * tan(27.7167°)<BR>\n" );
document.write( "* BD = 300 * tan(27.7167°)<BR>\n" );
document.write( "* BD ≈ 300 * 0.5244<BR>\n" );
document.write( "* BD ≈ 157.32 ft\r<BR>\n" );
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document.write( "**6. Calculate the Height of the Tower (CD):**\r<BR>\n" );
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document.write( "* CD = BD - BC<BR>\n" );
document.write( "* CD = 157.32 - 136.17<BR>\n" );
document.write( "* CD ≈ 21.15 ft\r<BR>\n" );
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document.write( "**7. Calculate the height of the mountain using the angles of elevation at B:**\r<BR>\n" );
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document.write( "* Let the height of the mountain be h.<BR>\n" );
document.write( "* tan(35°) = h/ distance from B to a point on the horizontal line from A.<BR>\n" );
document.write( "* Let the distance from B to a point on the horizontal line be x.<BR>\n" );
document.write( "* tan(39°) = (h+tower)/x.<BR>\n" );
document.write( "* tan(35°) = h/x<BR>\n" );
document.write( "* x = h/tan(35°)<BR>\n" );
document.write( "* tan(39°) = (h+tower)/(h/tan(35°))<BR>\n" );
document.write( "* tan(39°) = (h+tower)*tan(35°)/h<BR>\n" );
document.write( "* h*tan(39°) = h*tan(35°) + tower*tan(35°)<BR>\n" );
document.write( "* h(tan(39°)-tan(35°)) = tower*tan(35°)<BR>\n" );
document.write( "* tower = h(tan(39°)-tan(35°))/tan(35°)<BR>\n" );
document.write( "* tower = h(0.8098-0.7002)/0.7002<BR>\n" );
document.write( "* tower = h(0.1096)/0.7002<BR>\n" );
document.write( "* tower = 0.1565h.\r<BR>\n" );
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document.write( "**8. Calculate the height of the mountain using the first set of angles:**\r<BR>\n" );
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document.write( "* BC = 136.17 ft.\r<BR>\n" );
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document.write( "**9. Calculate the height of the tower using the first set of angles:**\r<BR>\n" );
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document.write( "* CD = 21.15 ft.\r<BR>\n" );
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document.write( "**10. Substitute BC into the equation for the tower:**\r<BR>\n" );
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document.write( "* tower = 0.1565*136.17<BR>\n" );
document.write( "* tower = 21.3193\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The height of the tower is approximately 21.15 feet.<BR>\n" );
document.write( " </SPAN>\n" );
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