document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168441>Question 1168441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two Coast Guard stations, A and B, are on an east-west line and are 115 km apart.   The bearing of a ship from station A is N 18o E, and the bearing of the same ship from station B is N 33o W.\r<BR>\n" );
document.write( "\n" );
document.write( "a)   How far is the ship from the east-west line connecting the two Coast Guard stations.<BR>\n" );
document.write( "b)   How far is the ship from station A?<BR>\n" );
document.write( "c)   How far is the ship from station B?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851568 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=851568\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Visualize the Situation**\r<BR>\n" );
document.write( "\n" );
document.write( "* Draw a horizontal line representing the east-west line connecting stations A and B.<BR>\n" );
document.write( "* Place A and B on this line, 115 km apart, with A to the west and B to the east.<BR>\n" );
document.write( "* Draw a line from A in the direction N 18° E.<BR>\n" );
document.write( "* Draw a line from B in the direction N 33° W.<BR>\n" );
document.write( "* The intersection of these two lines is the position of the ship (S).\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Set Up the Triangle**\r<BR>\n" );
document.write( "\n" );
document.write( "* Let S be the position of the ship.<BR>\n" );
document.write( "* We have triangle ABS.<BR>\n" );
document.write( "* AB = 115 km<BR>\n" );
document.write( "* Angle SAB = 90° - 18° = 72°<BR>\n" );
document.write( "* Angle SBA = 90° - 33° = 57°<BR>\n" );
document.write( "* Angle ASB = 180° - (72° + 57°) = 180° - 129° = 51°\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Use the Law of Sines**\r<BR>\n" );
document.write( "\n" );
document.write( "We can use the Law of Sines to find the distances AS and BS:\r<BR>\n" );
document.write( "\n" );
document.write( "* AS / sin(57°) = BS / sin(72°) = AB / sin(51°)\r<BR>\n" );
document.write( "\n" );
document.write( "**b) How far is the ship from station A? (AS)**\r<BR>\n" );
document.write( "\n" );
document.write( "* AS / sin(57°) = 115 / sin(51°)<BR>\n" );
document.write( "* AS = 115 * sin(57°) / sin(51°)<BR>\n" );
document.write( "* AS ≈ 115 * 0.8387 / 0.7771<BR>\n" );
document.write( "* AS ≈ 124.06 km\r<BR>\n" );
document.write( "\n" );
document.write( "**c) How far is the ship from station B? (BS)**\r<BR>\n" );
document.write( "\n" );
document.write( "* BS / sin(72°) = 115 / sin(51°)<BR>\n" );
document.write( "* BS = 115 * sin(72°) / sin(51°)<BR>\n" );
document.write( "* BS ≈ 115 * 0.9511 / 0.7771<BR>\n" );
document.write( "* BS ≈ 140.73 km\r<BR>\n" );
document.write( "\n" );
document.write( "**a) How far is the ship from the east-west line connecting the two Coast Guard stations?**\r<BR>\n" );
document.write( "\n" );
document.write( "Let's call the point where the perpendicular from S to AB meets AB as point P. We need to find the length of SP.\r<BR>\n" );
document.write( "\n" );
document.write( "We can use triangle ASP to find SP.\r<BR>\n" );
document.write( "\n" );
document.write( "* sin(72°) = SP / AS<BR>\n" );
document.write( "* SP = AS * sin(72°)<BR>\n" );
document.write( "* SP ≈ 124.06 * sin(72°)<BR>\n" );
document.write( "* SP ≈ 124.06 * 0.9511<BR>\n" );
document.write( "* SP ≈ 118 km\r<BR>\n" );
document.write( "\n" );
document.write( "Alternatively, we can use triangle BSP.\r<BR>\n" );
document.write( "\n" );
document.write( "* sin(57°) = SP / BS<BR>\n" );
document.write( "* SP = BS * sin(57°)<BR>\n" );
document.write( "* SP ≈ 140.73 * sin(57°)<BR>\n" );
document.write( "* SP ≈ 140.73 * 0.8387<BR>\n" );
document.write( "* SP ≈ 118 km\r<BR>\n" );
document.write( "\n" );
document.write( "**Answers:**\r<BR>\n" );
document.write( "\n" );
document.write( "a) The ship is approximately 118 km from the east-west line connecting the two Coast Guard stations.<BR>\n" );
document.write( "b) The ship is approximately 124.06 km from station A.<BR>\n" );
document.write( "c) The ship is approximately 140.73 km from station B.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );