document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173661>Question 1173661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Triangle A, B and C are three points on a horizontal field. A is due west of B, the bearing of B from C is 125 degrees, AB = 430m and BC = 460m.\r<BR>\n" );
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document.write( "At a certain instant, a hot air balloon is at a point which is directly above C.Given that the angle of elevation of the hot air balloon from B is 5.2 degrees, find the angle of elevation of the hot air balloon from A. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850721 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850721\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break this problem down step-by-step.\r<BR>\n" );
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document.write( "**1. Draw a Diagram**\r<BR>\n" );
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document.write( "* Draw a horizontal line segment AB, with A to the left of B.<BR>\n" );
document.write( "* Draw a line segment BC, where the bearing of B from C is 125 degrees. This means the angle between the north line from C and the line CB is 125 degrees. Since A is west of B, the angle ACB is 180 - 125 - 90 = 35 degrees.<BR>\n" );
document.write( "* Draw a vertical line segment from C upwards to represent the hot air balloon's position. Let's call this point D.<BR>\n" );
document.write( "* Connect BD and AD.\r<BR>\n" );
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document.write( "**2. Use the Law of Cosines to Find AC**\r<BR>\n" );
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document.write( "* We know AB = 430m, BC = 460m, and angle ACB = 35 degrees.<BR>\n" );
document.write( "* Use the law of cosines: AC² = AB² + BC² - 2(AB)(BC)cos(ACB)<BR>\n" );
document.write( "* AC² = 430² + 460² - 2(430)(460)cos(35°)<BR>\n" );
document.write( "* AC² = 184900 + 211600 - 395600cos(35°)<BR>\n" );
document.write( "* AC² = 396500 - 324484.7<BR>\n" );
document.write( "* AC² = 72015.3<BR>\n" );
document.write( "* AC = √72015.3 ≈ 268.36 m\r<BR>\n" );
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document.write( "**3. Find the Height of the Hot Air Balloon (CD)**\r<BR>\n" );
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document.write( "* The angle of elevation of the hot air balloon from B is 5.2 degrees.<BR>\n" );
document.write( "* We have a right triangle BCD.<BR>\n" );
document.write( "* tan(5.2°) = CD / BC<BR>\n" );
document.write( "* CD = BC * tan(5.2°)<BR>\n" );
document.write( "* CD = 460 * tan(5.2°)<BR>\n" );
document.write( "* CD ≈ 460 * 0.0909<BR>\n" );
document.write( "* CD ≈ 41.81 m\r<BR>\n" );
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document.write( "**4. Find the Angle of Elevation from A**\r<BR>\n" );
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document.write( "* We have a right triangle ACD.<BR>\n" );
document.write( "* We know AC ≈ 268.36 m and CD ≈ 41.81 m.<BR>\n" );
document.write( "* tan(angle CAD) = CD / AC<BR>\n" );
document.write( "* tan(angle CAD) = 41.81 / 268.36<BR>\n" );
document.write( "* tan(angle CAD) ≈ 0.1558<BR>\n" );
document.write( "* angle CAD = arctan(0.1558) ≈ 8.85 degrees\r<BR>\n" );
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document.write( "**Therefore, the angle of elevation of the hot air balloon from A is approximately 8.85 degrees.**<BR>\n" );
document.write( " </SPAN>\n" );
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