document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208283>Question 1208283</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ship sailing from a port P travels 50Km in a direction 070° to reach a port T. The ship then sails from T in a direction bearing 130° and a distance of 80km to a point Q. How far will the ship sail if it travels directly from P to Q? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846606 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=846606\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Exact Answer = <font color=red>10*sqrt(129) kilometers</font><BR>\n" );
document.write( "Approximate Answer = <font color=red>113.5782 kilometers</font><BR>\n" );
document.write( "Ask your teacher how s/he wants you to round the approximate value. \r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "The compass bearing has 000° pointing directly north. <BR>\n" );
document.write( "As you turn eastward, i.e. rotate clockwise, the angle increases.<BR>\n" );
document.write( "045° points to the northeast<BR>\n" );
document.write( "090° points east<BR>\n" );
document.write( "135° points southeast<BR>\n" );
document.write( "And so on.<BR>\n" );
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document.write( "With that in mind, here is what the diagram looks like<BR>\n" );
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document.write( "Angle APT = 70° and Angle BTQ = 130° are given<BR>\n" );
document.write( "Points A and B are directly north of P and T respectively. \r<BR>\n" );
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document.write( "Angle BTP = 110° is found by solving the equation angleAPT+angleBTP = 180. Note how vertical segments AP and BT are parallel, which means the consecutive interior angles are supplementary.<BR>\n" );
document.write( "Angle PTQ = 120° is determined by noting that the three angles around point T must add to 360 (you'll solve this equation: anglePTB+angleBTQ+anglePTQ = 360)\r<BR>\n" );
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document.write( "Focus on triangle PTQ.<BR>\n" );
document.write( "To find x, the length of segment PQ, we can use the Law of Cosines.<BR>\n" );
document.write( "c^2 = a^2 + b^2 - 2*a*b*cos(C)<BR>\n" );
document.write( "x^2 = 50^2 + 80^2 - 2*50*80*cos(120)<BR>\n" );
document.write( "x^2 = 50^2 + 80^2 - 2*50*80*(-1/2)<BR>\n" );
document.write( "x^2 = 12900<BR>\n" );
document.write( "x = sqrt(12900)<BR>\n" );
document.write( "x = sqrt(100*129)<BR>\n" );
document.write( "x = sqrt(100)*sqrt(129)<BR>\n" );
document.write( "x = <font color=red>10*sqrt(129) which is the exact distance</font><BR>\n" );
document.write( "x = 113.578166916005<BR>\n" );
document.write( "x = <font color=red>113.5782 kilometers which is the approximate distance</font><BR>\n" );
document.write( "Make sure that your calculator is set to degrees mode.<BR>\n" );
document.write( "Ask your teacher how s/he wants you to round this approximate value. \r<BR>\n" );
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document.write( "Another Approach.\r<BR>\n" );
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document.write( "Place P at the origin (0,0)<BR>\n" );
document.write( "This is how we'll locate point T.<BR>\n" );
document.write( "T = P + 50*( sin(70), cos(70) )<BR>\n" );
document.write( "T = (0,0) + 50*( 0.9396926, 0.3420201 )<BR>\n" );
document.write( "T = ( 50*0.9396926, 50*0.3420201 )<BR>\n" );
document.write( "T = ( 46.98463, 17.101005 )<BR>\n" );
document.write( "The decimal values are approximate. <BR>\n" );
document.write( "Make sure that your calculator is set to degrees mode.\r<BR>\n" );
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document.write( "Normally cosine is associated with the x coordinate; however, the compass bearing angles have 000° pointing north (rather than east), so we have a 90° rotation. This 90° rotation swaps the roles of sine and cosine. Cosine is a 90° phase-shifted version of sine.\r<BR>\n" );
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document.write( "We'll follow a similar method to find where point Q is located.<BR>\n" );
document.write( "Q = T + 80*( sin(130), cos(130) )<BR>\n" );
document.write( "Q = ( 46.98463, 17.101005 ) + 80*( 0.7660444, -0.6427876 )<BR>\n" );
document.write( "Q = ( 46.98463, 17.101005 ) + ( 80*0.7660444, 80*(-0.6427876) )<BR>\n" );
document.write( "Q = ( 46.98463, 17.101005 ) + ( 61.283552, -51.423008 )<BR>\n" );
document.write( "Q = ( 46.98463+61.283552, 17.101005+(-51.423008) )<BR>\n" );
document.write( "Q = ( 108.268182, -34.322003 )<BR>\n" );
document.write( "The decimal values are approximate. \r<BR>\n" );
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document.write( "Here is the calculation template for point Q's coordinates in one single line<BR>\n" );
document.write( "Q = ( 50*sin(70)+80*sin(130), 50*cos(70)+80*cos(130) )<BR>\n" );
document.write( "That line is based off of this<BR>\n" );
document.write( "Q = P + 50*( sin(70), cos(70) ) + 80*( sin(130), cos(130) )\r<BR>\n" );
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document.write( "The key takeaways are these locations<BR>\n" );
document.write( "P = (0,0)<BR>\n" );
document.write( "Q = ( 108.268182, -34.322003 ) which is approximate\r<BR>\n" );
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document.write( "Use the distance formula to find that<BR>\n" );
document.write( "PQ = sqrt( (108.268182)^2 + (-34.322003)^2 ) = <font color=red>113.5782 kilometers approximately</font><BR>\n" );
document.write( "The answer will vary depending on the rounding precision.\r<BR>\n" );
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document.write( "More practice with similar questions<BR>\n" );
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document.write( "and<BR>\n" );
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