document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151489>Question 1151489</A>: <I><SPAN STYLE=\"word-break: break-all;\"> during a portage, the leader of a scout troop decided to save time by hiking around a wooded area between the base camp and a lake. the troop reached the lake by walking 30 min on a bearing of N60E , then for 45 minutes on a bearing of N80W. the leader figures that they walked at an average speed of 3km/h<BR>\n" );
document.write( "Draw a diagram showing the information given<BR>\n" );
document.write( "If they could have walked straight through the forest area at 1km/h , did the troop save any time by going around ? </SPAN>\n" );
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document.write( "The instructions give these facts<BR>\n" );
document.write( "<ul><li>Fact 1: They walk for 30 minutes (0.5 hrs) on a bearing of N60E</li><li>Fact 2: They walk for 45 minutes (0.75 hrs) on a bearing of N80W</li><li>Fact 3: The average walking speed is 3 km/h</li></ul>\r<BR>\n" );
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document.write( "Let's set up the diagram. <BR>\n" );
document.write( "Step 1) Plot point A as the starting point. Draw a faint dashed line segment straight north of this point to set up a guideline. Then draw a 60 degree angle off this vertical segment to represent the bearing of N60E. This is the left most red angle in the diagram below. This is angle DAB.<BR>\n" );
document.write( "For more information about compass bearings, see this page below.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html\">http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html</a>\r<BR>\n" );
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document.write( "Step 2) Following this bearing, draw a line segment and mark it to be 1.5 km. This value comes from fact 1 and fact 3. Distance = rate*time = 3*0.5 = 1.5; Mark point B at the other end of the segment.\r<BR>\n" );
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document.write( "Step 3) Draw a vertical dashed line through point B. Make sure to extend the line above and below point B. Plot points E and F at the endpoints of this dashed segment. I have point E north of B, while F is south of B. Angle ABF is 60 degrees because EF is parallel to AD, also because angles DAB and ABF are alternate interior angles.\r<BR>\n" );
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document.write( "Step 4) Mark angle EBC to be 80 degrees. With point B as the reference point, look directly north to point E, then turn 80 degrees to the west to look at point C. This is the bearing N80W.\r<BR>\n" );
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document.write( "Step 5) Walk 2.25 km from B to C. The 2.25 comes from fact 2 and fact 3. Distance = rate*time = 3*0.75 = 2.25\r<BR>\n" );
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document.write( "Step 6) The red and blue angles will help us find the green angle (see diagram below). The three angles add to 180. If x is the green angle, then 80+x+60 = 180 solves to x = 40. Therefore, angle ABC is 40 degrees, which is the green angle.\r<BR>\n" );
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document.write( "After those steps, you should have this triangle set up<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/UcPO5w0.png\">\r<BR>\n" );
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document.write( "Let's clean up a bit of the extra unneeded segments and points. <BR>\n" );
document.write( "We also don't need the compass anymore.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/S7mc4ip.png\">\r<BR>\n" );
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document.write( "Sides:<BR>\n" );
document.write( "a = 2.25 km<BR>\n" );
document.write( "b = unknown<BR>\n" );
document.write( "c = 1.5 km\r<BR>\n" );
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document.write( "Angles<BR>\n" );
document.write( "A = unknown<BR>\n" );
document.write( "B = 40 degrees<BR>\n" );
document.write( "C = unknown\r<BR>\n" );
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document.write( "Use the Law of Cosines to solve for side 'b'<BR>\n" );
document.write( "b^2 = a^2 + c^2 - 2*a*c*cos(B)<BR>\n" );
document.write( "b^2 = 2.25^2 + 1.5^2 - 2*2.25*1.5*cos(40)<BR>\n" );
document.write( "b^2 = 5.0625 + 2.25 - 6.75*cos(40)<BR>\n" );
document.write( "b^2 = 5.0625 + 2.25 - 6.75*0.766044443118978<BR>\n" );
document.write( "b^2 = 5.0625 + 2.25 - 5.1707999910531<BR>\n" );
document.write( "b^2 = 2.1417000089469<BR>\n" );
document.write( "b = sqrt(2.1417000089469)<BR>\n" );
document.write( "b = 1.46345481957828<BR>\n" );
document.write( "b = 1.46345\r<BR>\n" );
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document.write( "Side b is roughly 1.46345 km<BR>\n" );
document.write( "So the distance from A to C is roughly 1.46345 km.<BR>\n" );
document.write( "Walking at a speed of 1 km/hr through the forest, it will take roughly t = d/r = 1.46345/1 = 1.46345 hours to go from A to C. \r<BR>\n" );
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document.write( "If we went from A to B then to C, then it would take<BR>\n" );
document.write( "(30 min)+(45 min) = (0.5 hr)+(0.75 hr) = 1.25 hr\r<BR>\n" );
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document.write( "The time difference is:<BR>\n" );
document.write( "1.46345 - 1.25 = 0.21345 hours\r<BR>\n" );
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document.write( "0.21345 hours = (0.21345 hours)*(60 min/1 hr)<BR>\n" );
document.write( "0.21345 hours = 12.807 minutes <BR>\n" );
document.write( "12.807 minutes = 12 min + 0.807 min<BR>\n" );
document.write( "12.807 minutes = 12 min + (0.807 min)*(60 sec/1 min)<BR>\n" );
document.write( "12.807 minutes = 12 min + 48.42 seconds<BR>\n" );
document.write( "12.807 minutes = 12 min, 48.42 seconds\r<BR>\n" );
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document.write( "---------------------------------------------------------------\r<BR>\n" );
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document.write( "Answer: Yes the troop saved time by going around the forest. <BR>\n" );
document.write( "They saved about 12.807 minutes (equivalent to 12 min, 48.42 seconds).\r<BR>\n" );
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