document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083446>Question 1083446</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two ships sail out from a harbour at the same time. One sails on a bearing of 015 degrees and travels a distance of 32 miles. The other ship sails 47 miles on a bearing of 165 degrees.\r<BR>\n" );
document.write( "\n" );
document.write( "Draw a picture and<BR>\n" );
document.write( "a) How far apart are the ships?<BR>\n" );
document.write( "b) What is the bearing from the first ship to the second ship?\r<BR>\n" );
document.write( "\n" );
document.write( "Thank You </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697441 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=697441\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" size=\"4\"><BR>\n" );
document.write( "Answer to part a): <font color=red>76.4068348682472 miles</font><BR>\n" );
document.write( "Answer to part b): <font color=red>177 degrees</font></font><BR>\n" );
document.write( "<font color=\"black\" face=\"times\" size=\"4\"><BR>\n" );
document.write( "Note: The answer for part a) is approximate. Round this value to whatever you need to. <BR>\n" );
document.write( "-------------------------------------------------------------------------------------------------<BR>\n" );
document.write( "-------------------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First set up the drawing\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 1) Plot the harbor (point H). Then draw a straight line pointing directly north to point N. This will represent true north on the compass. The bearing angles will originate from this direction\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 2) Face true north and rotate 15 degrees clockwise as shown by the red angle. Draw a segment that goes from point H to point A. Label this segment 32 (to represent ship A has traveled 32 miles). So AH = 32. Angle NHA = 15 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 3) Turn back to face true north again. Draw a segment HB such that angle NHB is 165 degrees as shown by the blue angle. Denote HB = 47 to indicate that ship B has traveled 47 miles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 4) Draw a segment from A to B. The goal is to find the length of this segment. It's unknown for now, so we'll call it x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "After those steps are completed, you should get something like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/Xz8RA5A.png\"><BR>\n" );
document.write( "The summary information of that drawing is<BR>\n" );
document.write( "point H = harbor<BR>\n" );
document.write( "point A = ship A's location (after traveling 32 miles from the harbor)<BR>\n" );
document.write( "point B = ship B's location (after traveling 47 miles from the harbor)<BR>\n" );
document.write( "point N = true north<BR>\n" );
document.write( "--------<BR>\n" );
document.write( "segment HA = 32 miles<BR>\n" );
document.write( "segment HB = 47 miles<BR>\n" );
document.write( "segment AB = x miles<BR>\n" );
document.write( "--------<BR>\n" );
document.write( "angle NHA = 15 degrees (red angle)<BR>\n" );
document.write( "angle NHB = 165 degrees (blue angle)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can use the law of cosines to find the value of x. But first we need to figure out what angle AHB is equal to. The angle addition postulate says\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(angle AHB) + (angle NHA) = (angle NHB)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug in the known angles and isolate \"angle AHB\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(angle AHB) + (angle NHA) = (angle NHB)<BR>\n" );
document.write( "(angle AHB) + (15) = (165)<BR>\n" );
document.write( "(angle AHB) + 15 = 165<BR>\n" );
document.write( "(angle AHB) + 15-15 = 165-15<BR>\n" );
document.write( "angle AHB = 150\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's update the drawing. I'm going to erase the other two angles (15 and 165 degrees) since they won't be needed further. In their place, I'll add in angle AHB = 150 degrees. I'll also erase point N and the arrow pointing north since we don't need that anymore either.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We now have this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"https://i.imgur.com/QeLBzF3.png\"><BR>\n" );
document.write( "The angle C = 150 is between the two sides of a = 32 and b = 47 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Using the law of cosines, we can say<BR>\n" );
document.write( "c^2 = a^2 + b^2 - 2*a*b*cos(C)<BR>\n" );
document.write( "x^2 = 32^2 + 47^2 - 2*32*47*cos(150)<BR>\n" );
document.write( "x^2 = 5838.0044145836<BR>\n" );
document.write( "sqrt(x^2) = sqrt(5838.0044145836)<BR>\n" );
document.write( "x = 76.4068348682472\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore the distance from A to B is approximately 76.4068348682472 miles. Round this value to whatever you need to or however the teacher requires. This is the answer to part A\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To get the bearing from the first ship to the second ship, we need to add on a bit to the drawing. Let's draw a segment that goes straight up from point A. Plot C on this segment. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Also, extend out segment HA (as shown by the dashed black line) and plot D on this dashed segment. \r<BR>\n" );
document.write( "\n" );
document.write( "I'll mark the bearing as the angle y. It's unknown for now.<BR>\n" );
document.write( "Angle z = angle DAB\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/BdacYcH.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "AC || NH, so this means that <BR>\n" );
document.write( "angle CAD = angle NHA = 15 degrees\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the law of cosines to find angle A\r<BR>\n" );
document.write( "\n" );
document.write( "sin(A)/a = sin(H)/h<BR>\n" );
document.write( "sin(A)/47 = sin(150)/x<BR>\n" );
document.write( "sin(A)/47 = sin(150)/76.4068348682472<BR>\n" );
document.write( "sin(A) = 47*sin(150)/76.4068348682472<BR>\n" );
document.write( "sin(A) = 0.30756410785138<BR>\n" );
document.write( "A = arcsin(0.30756410785138)<BR>\n" );
document.write( "A = 17.912493477208\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this means<BR>\n" );
document.write( "angle NAB = 17.912493477208 degrees\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "angle DAB = 180 - (angle NAB)<BR>\n" );
document.write( "angle DAB = 180 - (17.912493477208)<BR>\n" );
document.write( "angle DAB = 162.087506522792\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "angle CAB = (angle CAD) + (angle DAB)<BR>\n" );
document.write( "angle CAB = (15) + (162.087506522792)<BR>\n" );
document.write( "angle CAB = 177.087506522792<BR>\n" );
document.write( "which rounds to 177 degrees\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so y = 177 roughly<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );