document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=335832>Question 335832</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two ships P and Q leaves a point at the same tine . P sails at 10km/hr  on a bearing of 030 degree and Q sails at 12km/hr on a bearing of 300 degree calculate their distances apart and the bearing of P from Q after 2 hours </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #240837 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=240837\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> P sails at 10 km/hr on a bearing of 30 degrees.<BR>\n" );
document.write( "Q sails at 12 km/hr on a bearing of 60 degrees.\r<BR>\n" );
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document.write( "Plot their vectors on a grid starting from the center, and you will see that their combined vectors form a right triangle that has one leg with a length of 10 kilometers per hour, and another leg with a length of 12 kilometers per hour.\r<BR>\n" );
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document.write( "If you mark the center of the grid as A, then P goes from A to B forming a 30 degree angles with the horizontal axis, and Q goes from A to C forming a negative 60 degrees with the horizontal axis.\r<BR>\n" );
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document.write( "The triangle formed is ABC, where the 90 degree angle is at A (30 degrees plus 60 degrees = 90 degrees).\r<BR>\n" );
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document.write( "Side a is opposite angle A and is formed by the line segment BC.<BR>\n" );
document.write( "Side b is opposite angle B and is formed by the line segment AC.<BR>\n" );
document.write( "Side c is opposite angle C and is formed by he line segment AB.\r<BR>\n" );
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document.write( "Since the 90 degree angle is at angle A, then the hypotenuse of this right triangle is side a which is formed by the line segment BC.\r<BR>\n" );
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document.write( "Since this is a right triangle, you can find the hypotenuse of this triangle by using the Pythagorean theorem of a^2 + b^2 = c^2.\r<BR>\n" );
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document.write( "You get 10^2 + 12^2 = 100 + 144 = 244.\r<BR>\n" );
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document.write( "Square root of 244 is equal to 15.62049935.\r<BR>\n" );
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document.write( "This means that the points B and C are increasing the distance between them at a rate of 15.62049935 kilometers per hour.\r<BR>\n" );
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document.write( "In 2 hours, this means that the ships are 31.2409987 miles apart.\r<BR>\n" );
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document.write( "In 4 hours, the ships will be 62.48199741 miles apart, etc.\r<BR>\n" );
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document.write( "If the angle formed by them was not 90 degrees, then you could have found the third side using the Law of Cosines formula of:\r<BR>\n" );
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document.write( "a^2 = b^2 + c^2 - 2*b*c*cos(A).\r<BR>\n" );
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document.write( "With this formula, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.\r<BR>\n" );
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document.write( "As long as their velocity remains the same, the distance between them will continue to increase by that velocity multiplied by the time in hours.\r<BR>\n" );
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document.write( "You can see this better by creating a distance vector rather than a velocity vector.\r<BR>\n" );
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document.write( "The distance vector depends on time.\r<BR>\n" );
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document.write( "In 1 hour, P travels a distance of 10 kilometers at a bearing of 30 degrees and Q travels a distance of 12 kilometers at a bearing of 300 degrees (-60 degrees).\r<BR>\n" );
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document.write( "In 2 hours, P travels a distance of 20 kilometers at a bearing of 30 degrees and Q travels a distance of 24 kilometers as a bearing of 300 degrees.\r<BR>\n" );
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document.write( "The triangle formed for the 2 hour distances is still a right triangle labeled ABC where A is the origin of both boats, and B is the point that P has reached, and C is the point that Q has reached.  Triangle ABC is therefore labeled with a 90 degree at angle A, and side AB = 20 kilometers, and side AC = 24 kilometers.\r<BR>\n" );
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document.write( "Since side AB is opposite angle C, then side AB is called side c.\r<BR>\n" );
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document.write( "Since side AC is opposite angle B, then side AC is called side b.\r<BR>\n" );
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document.write( "The side opposite angle A is called side a which is formed by the line segment BC.\r<BR>\n" );
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document.write( "You want to find the length of side a because it is the distance between the 2 boats.\r<BR>\n" );
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document.write( "Using the right triangle formula and since side a is the hypotenuse of the right triangle formed, you get a^2 = b^2 + c^2 which becomes a^2 = 20^2 + 24^2 = 31.2409987 kilometers.\r<BR>\n" );
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document.write( "This is exactly what we derived above.\r<BR>\n" );
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document.write( "Using the Law of Cosines, the formula becomes:\r<BR>\n" );
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document.write( "a^2 = b^2 + c^2 - 2*b*c*cos(C).\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "a^2 = 20^2 + 24^2 - 2*20*24*cos(90).\r<BR>\n" );
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document.write( "Since cos(90) = 0, this formula becomes:\r<BR>\n" );
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document.write( "a^2 = 20^2 + 24^2 which is, once again, exactly what we derived above.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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