document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189731>Question 1189731</A>: <I><SPAN STYLE=\"word-break: break-all;\"> While standing on a 75m tall bridge, given can see two boats. From his position on the bridge, the first boat is located on a bearing of 70 degrees and the second boat is located on a bearing of 300 degrees. Gavin estimates that the angles of depression for each of the boats are 38 degrees and 47 degrees respectively. How far apart are the boats?\r<BR>\n" );
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document.write( "Answer: <font color=red>150.79 meters approximately</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Define these points<ul><li>A = location of boat 1</li><li>B = location of boat 2</li><li>C = point on the water's surface directly below Gavin's location</li><li>D = Gavin's location on the bridge</li></ul>For the sake of simplicity, we'll assume points A,B,C are all on the same flat plane.<BR>\n" );
document.write( "Realistically, the water is bobbing the boats around, so such an assumption isn't entirely valid (but again we'll make things relatively simple).\r<BR>\n" );
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document.write( "Due to the 3D nature of this problem, we'll have to break it down into 2D pieces.\r<BR>\n" );
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document.write( "On a separate part of the paper, draw out right triangle ACD. This is a side profile view for the boat 1.\r<BR>\n" );
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document.write( "Here are all the relevant drawings needed.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"75%\" src = \"https://i.imgur.com/YCCT5Fl.png\"><BR>\n" );
document.write( "For now, focus on triangle ACD only.<BR>\n" );
document.write( "That triangle has these sides<BR>\n" );
document.write( "CD = 75 meters = height<BR>\n" );
document.write( "AC = unknown\r<BR>\n" );
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document.write( "Use the tangent ratio to get...<BR>\n" );
document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(A) = CD/AC<BR>\n" );
document.write( "tan(38) = 75/AC<BR>\n" );
document.write( "AC*tan(38) = 75<BR>\n" );
document.write( "AC = 75/tan(38)<BR>\n" );
document.write( "AC = 95.99562 approximately<BR>\n" );
document.write( "Notice how angle A of triangle ACD is equal to the angle of depression for the first boat.<BR>\n" );
document.write( "This is due to the alternate interior angles theorem.\r<BR>\n" );
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document.write( "Through similar calculations for triangle BCD, you should find that BC = 75/tan(47) = 69.93863 meters approximately.\r<BR>\n" );
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document.write( "Now we'll consider the birds-eye-view to look directly down on triangle ABC. \r<BR>\n" );
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document.write( "We already found that<BR>\n" );
document.write( "AC = 95.99562<BR>\n" );
document.write( "BC = 69.93863<BR>\n" );
document.write( "which are sides b and 'a' in that order<BR>\n" );
document.write( "In other words,<BR>\n" );
document.write( "b = AC = 95.99562<BR>\n" );
document.write( "a = BC = 69.93863\r<BR>\n" );
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document.write( "The ultimate goal is to find the length of segment c = AB, which is the distance between the boats.\r<BR>\n" );
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document.write( "The bearings 70 and 300 have a gap of 300-70 = 230 degrees between them.<BR>\n" );
document.write( "The remaining bit is 360-230 = 130 degrees<BR>\n" );
document.write( "This measures angle ACB, aka angle C.\r<BR>\n" );
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document.write( "Use the law of cosines to find side c = AB<BR>\n" );
document.write( "c^2 = a^2+b^2-2*a*b*cos(C)<BR>\n" );
document.write( "c^2 = 69.93863^2+95.99562^2-2*69.93863*95.99562*cos(130)<BR>\n" );
document.write( "c^2 = 22,737.6687<BR>\n" );
document.write( "c = sqrt(22,737.6687)<BR>\n" );
document.write( "c = <font color=red>150.79 meters is the approximate distance between the two boats</font><BR>\n" );
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