document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1113500>Question 1113500</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help. Thank you. <BR>\n" );
document.write( "A ship leaves port and travels for 50 km along a bearing of N 25 degree W, then turns along a bearing of S 70 degree W for 100 km, followed by a 75 km leg along a bearing of S 20 degree E, and finally along a bearing of N 80 degree E for 80 km. <BR>\n" );
document.write( "a) where is ship's position relative to the port?<BR>\n" );
document.write( "b) How far is the ship from the port?<BR>\n" );
document.write( "c) Along what bearing must the ship travel to return to port? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #728595 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=728595\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the concept here:\r<BR>\n" );
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document.write( "add up all the north direction distances and subtract all the south direction distances to get a net distance that is either north or south.\r<BR>\n" );
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document.write( "add up all the east direction distances and subtract all the west direction distances to get a net distance that is either east or west.\r<BR>\n" );
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document.write( "the ship starts at point A.\r<BR>\n" );
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document.write( "that would be the origin on the graph.\r<BR>\n" );
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document.write( "it travels 50 kilometers to point B on a bearing of north 25 degrees west.\r<BR>\n" );
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document.write( "this forms right triangle BAH where the north distance is 50 * cos(25) and the west distance is 50 * sin(25).\r<BR>\n" );
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document.write( "the north distance is equal to AH which is equal to 50 * cos(25) which is equal to 45.31538935 kilometers.\r<BR>\n" );
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document.write( "the west distance is equal to HB which is equal to 50 * sin(25) which is equal to 21.13091309 kilometers.\r<BR>\n" );
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document.write( "what that looks like is in the following graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/033002.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "from point B, 100 kilometers on a bearing of south 70 degrees west.\r<BR>\n" );
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document.write( "this forms right triangle BCI where the south distance is 100 * cos(70) and the west distance is 100 * sin(70).\r<BR>\n" );
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document.write( "the south distance is equal to BI which is equal to 100 * cos(70) which is equal to 34.20201433 kilometers.\r<BR>\n" );
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document.write( "the west distance is equal to IC which is equal to 100 * sin(70) which is equal to 93.96926208 kilometers.\r<BR>\n" );
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document.write( "what that looks like is in the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/033003.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "from point C, he travels 75 kilometers on a bearing of south 20 degrees east.\r<BR>\n" );
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document.write( "this forms right triangle CDJ where the south distance is 75 * cos(20) and the east distance is 75 * sin(30).\r<BR>\n" );
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document.write( "the south distance is equal to CJ which is equal to 75 * cos(20) which is equal to 70.47694656 kilometers.\r<BR>\n" );
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document.write( "the east distance is equal to JD which is equal to 75 * cos(20) which is equal to 25.65151075 kilometers.\r<BR>\n" );
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document.write( "what that looks like is in the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/033004.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "from point D, he travels 80 kilometers to point F on a bearing of north 80 degrees east.\r<BR>\n" );
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document.write( "this forms right triangle DFK where the north distance is 80 * cos(80) and the east distance is 80 * sin(80).\r<BR>\n" );
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document.write( "the north distance is equal to DK which is equal to 80 * cos(80) which is equal to 13.89185421 kilometers.\r<BR>\n" );
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document.write( "the east distance is equal to 80 * sin(80) which is equal to KF which is equal to 78.78462024 kilometers.\r<BR>\n" );
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document.write( "what this looks like is in the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/033005.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "if we consider the distance going north to be positive and the distance going south to be negative, we get a net distance of 50 * cos(25) km - 100 * cos(70) km - 75 * cos(20) km + 80 * cos(80) which is equal to -45.47171733.\r<BR>\n" );
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document.write( "this means he has traveled a net distance of 45.47171733 kilometers directly south of his home port.\r<BR>\n" );
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document.write( "if we consider the distance going west to be negative and the distance going east to be positive, we get a net distance of -50 * sin(25) - 100 * sin(70) + 75 * sin(20) + 80 * sin(80) which is equal to -10.66404418.\r<BR>\n" );
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document.write( "this means he has traveled a net distance of 10.66404418 kilometers directly west of his home port.\r<BR>\n" );
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document.write( "this forms right triangle FLA which is our final triangle.\r<BR>\n" );
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document.write( "the distance from point F to L is equal to FL which is equal to 45.47171733 kilometers.\r<BR>\n" );
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document.write( "the distance from point L to point A is equal to LA which is equal to 10.66404418 kilometers.\r<BR>\n" );
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document.write( "the distance from point A to point F is equal to FA which is the hypotenuse of right triangle FAL and is therefore equal to square root of ((45.47171733)^2 + (10.66404418)^2) which is equal to 46.70544845 kilometers.\r<BR>\n" );
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document.write( "the net bearing from point F to point A would be arc tangent (LA / FL) which is equal to arc tangent (10.66404418 / 45.47171733) which is equal to north 13.19850608 degrees east.\r<BR>\n" );
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document.write( "what this looks like is shown in the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/033006.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the answers to your questions are, as best i can determine, are shown below:\r<BR>\n" );
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document.write( "a) where is ship's position relative to the port?\r<BR>\n" );
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document.write( "the ship is 45.47171733 kilometers south of the home port and 10.66404418 kilometers west of the home port.\r<BR>\n" );
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document.write( "b) How far is the ship from the port?\r<BR>\n" );
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document.write( "the ship is 46.70544845 kilometers from the home port as measured by the straight line distance between the final destination of the ship and the home port.\r<BR>\n" );
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document.write( "c) Along what bearing must the ship travel to return to port?\r<BR>\n" );
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document.write( "the ship would have to travel on a bearing of north 13.19850608 degrees east to get back to the home port from its final destination.\r<BR>\n" );
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document.write( "the overall graph of the ship's route is shown in the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/033007.jpg\" alt=\"$$$\" > </SPAN>\n" );
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