document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203473>Question 1203473</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A plane flies on a true bearing of 320° for 450 km. It then flies on a true bearing of 350° for 130 km and finally on a true bearing of 050° for 330 km. How far north of its starting point is the plane?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839077 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839077\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The answer is approximately <font color=red>684.864918 km</font>. I rounded to 6 decimal places. Feel free to round however else your teacher instructs. I have confirmed this answer with GeoGebra.\r<BR>\n" );
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document.write( "Tutor ikleyn has a value (684.864691) close to what I got. However there's slight rounding error. The \"684.864\" portions match up at least.\r<BR>\n" );
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document.write( "Tutor theo has at least one error in his calculations. The good news is that 687.4045186 is somewhat close to 684.864918\r<BR>\n" );
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document.write( "I'll explain why the formula ikleyn uses works. \r<BR>\n" );
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document.write( "The four key bearing angles to memorize are:<BR>\n" );
document.write( "000° = north<BR>\n" );
document.write( "090° = east<BR>\n" );
document.write( "180° = south<BR>\n" );
document.write( "270° = west<BR>\n" );
document.write( "Check out the diagram below. <BR>\n" );
document.write( "Basically we start aiming north. Then rotating clockwise will increase the bearing angle. \r<BR>\n" );
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document.write( "So let's say the bearing is 050° and we move 100 km along this bearing. <BR>\n" );
document.write( "We move 100 km along the red arrow.<BR>\n" );
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document.write( "We want to know how far north we are from the origin. <BR>\n" );
document.write( "Thus we want to find the vertical leg of this right triangle marked in red.<BR>\n" );
document.write( "cos(angle) = adjacent/hypotenuse<BR>\n" );
document.write( "cos(50) = adjacent/100<BR>\n" );
document.write( "adjacent = 100*cos(50)\r<BR>\n" );
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document.write( "Therefore the north-south displacement for this example is 100*cos(50) = 64.27876 km approximately.\r<BR>\n" );
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document.write( "In general if you move r units along bearing theta degrees, then r*cos(theta) units is the north-south displacement. <BR>\n" );
document.write( "Negative displacement means we move south, while positive displacements move us north.\r<BR>\n" );
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document.write( "This idea can then be applied many times to chain together multiple movements. <BR>\n" );
document.write( "That is how ikleyn ended up with the formula: 450*cos(320°) + 130*cos(350°) + 330*cos(50°) <BR>\n" );
document.write( "Make sure your calculator is in degree mode. A quick check could be something like cos(60°) = 0.5 which you should have memorized.\r<BR>\n" );
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document.write( "Feel free to ask any further questions if you're still stuck.<BR>\n" );
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