document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208644>Question 1208644</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Royal Cruise and Carnival Cruise start at island point A. Royal sails 35 km on a bearing N 40 E to island B. Carnival sails in the direction S 30 W to island C. The distance from A to C is 70 km. How far is it from B to C? </SPAN>\n" );
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document.write( "Answer: <font color=red>104.644914 km</font> (approximate)\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Draw an xy axis. <BR>\n" );
document.write( "Let's place point A at the origin (0,0).\r<BR>\n" );
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document.write( "Island B is somewhere in the northeast quadrant (aka upper right corner). <BR>\n" );
document.write( "This is because notation like \"N 40 E\" means we face directly north and then turn 40 degrees toward the east.<BR>\n" );
document.write( "Draw a segment to connect points A and B. <BR>\n" );
document.write( "This segment will be the hypotenuse of a right triangle such that its vertical leg is on top of the positive y axis. <BR>\n" );
document.write( "The horizontal leg is <font size=4>not</font> on the x axis. Refer to the diagram shown below.<BR>\n" );
document.write( "The vertical leg is labeled as dy.<BR>\n" );
document.write( "The horizontal leg is labeled as dx.<BR>\n" );
document.write( "The angle theta (symbol <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is the acute angle between the hypotenuse and vertical leg.<BR>\n" );
document.write( "r = hypotenuse = distance the boat travels from A to B<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Acircle%280%2C0%2C0.05%29%2Ccircle%280%2C0%2C0.07%29%2Ccircle%280%2C0%2C0.09%29%2Ccircle%280%2C0%2C0.11%29%2Ccircle%284%2C4%2C0.05%29%2Ccircle%284%2C4%2C0.07%29%2Ccircle%284%2C4%2C0.09%29%2Ccircle%284%2C4%2C0.11%29%2C%0D%0Aline%280%2C0%2C4%2C4%29%2Cline%284%2C4%2C0%2C4%29%2Cline%280%2C4%2C0%2C0%29%2C%0D%0Aline%280%2C3.7%2C0.3%2C3.7%29%2Cline%280.3%2C3.7%2C0.3%2C4%29%2C%0D%0Aline%28-30%2C0%2C30%2C0%29%2Cline%280%2C30%2C0%2C-30%29%2C%0D%0Alocate%280.2%2C-0.2%2C%22A%22%29%2Clocate%284.2%2C3.8%2C%22B%22%29%2Clocate%281.86%2C4.4%2C%22dx%22%29%2Clocate%28-0.62%2C2.26%2C%22dy%22%29%2Clocate%282.38%2C1.74%2C%22r%22%29%2Clocate%280.46-0.2%2C1.7-0.4%2Ctheta%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If you know your trig ratios, then you should realize that:<BR>\n" );
document.write( "<font color=blue>dx = r*sin(theta)<BR>\n" );
document.write( "dy = r*cos(theta)</font>\r<BR>\n" );
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document.write( "Those formulas are very useful to determine the location of island B when A = (0,0).<BR>\n" );
document.write( "The dx and dy are the components of the translation vector to get from A to B.<BR>\n" );
document.write( "You should find that:<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">dx = r*sin(theta) = 35*sin(40) = 22.497566\r\n" );
document.write( "dy = r*cos(theta) = 35*cos(40) = 26.811556</PRE>Each result is approximate. Your calculator needs to be in degree mode.<BR>\n" );
document.write( "Point B is at the approximate location B = (22.497566, 26.811556) when A = (0,0).\r<BR>\n" );
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document.write( "We'll use the same idea, more or less, to figure out where point C is located on the xy grid.<BR>\n" );
document.write( "Let's calculate the dx and dy values when the Carnival boat sails r = 70 kilometers and the angle mentioned is theta = 30. <BR>\n" );
document.write( "We'll worry about the southwest direction later.<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">dx = r*sin(theta) = 70*sin(30) = 35\r\n" );
document.write( "dy = r*cos(theta) = 70*cos(30) = 60.621778 approximately</PRE>Now if Carnival was sailing northeast, then we wouldn't make any adjustments to the signs of dx and dy.\r<BR>\n" );
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document.write( "However, it's instead going southwest, so we need to flip the sign of each.<BR>\n" );
document.write( "The location of island C is roughly  (-35, -60.621778) when A = (0,0).\r<BR>\n" );
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document.write( "I suppose you could have this rough sketch on a notecard (or have it memorized)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C300%2C-5%2C5%2C-5%2C5%2C%0D%0Aline%28-20%2C0%2C20%2C0%29%2Cline%280%2C-20%2C0%2C20%29%2C%0D%0Alocate%282%2C2%2C%22%28dx%2Cdy%29%22%29%2Clocate%282%2C-2%2C%22%28dx%2C-dy%29%22%29%2Clocate%28-3%2C-2%2C%22%28-dx%2C-%0D%0Ady%29%22%29%2Clocate%28-3%2C2%2C%22%28-dx%2Cdy%29%22%29%2C%0D%0Alocate%282%2C2.5%2C%22Northeast%22%29%2Clocate%282%2C-2.5%2C%22Southeast%22%29%2Clocate%28-3%2C-2.5%2C%22Southwest%22%29%2Clocate%28-3%2C2.5%2C%22Northwest%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Or just remember to use common sense intuition when figuring out the positive or negative signs.\r<BR>\n" );
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document.write( "Summary so far:<BR>\n" );
document.write( "Islands B and C are at these approximate locations when A = (0,0)<ul><li>B = (22.497566, 26.811556)</li><li>C = (-35, -60.621778)</li></ul>The next step from here is to use the distance formula to figure out how far it is from B to C.<BR>\n" );
document.write( "Distance formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d+=+sqrt%28+%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2+%29+ ALIGN=MIDDLE  ALT=\"+d+=+sqrt%28+%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll let the student do the scratch work. <BR>\n" );
document.write( "You should get the <font color=red>approximate answer of 104.644914 km.</font><BR>\n" );
document.write( "The answer will slightly vary depending how your teacher wants you to round. \r<BR>\n" );
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document.write( "I used <a  rel=nofollow HREF=\"https://www.geogebra.org/\">GeoGebra</a> to verify the answer.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.geogebra.org/m/nut69y8e\">Here</a> is the specific worksheet I made for this problem. If the page is blank, then hover your mouse over any part of the box. A spiral arrow should show up. Click on it to have it refresh the page. <BR>\n" );
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