document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204688>Question 1204688</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rocco and Biff are two koala bears frolicking in a meadow. Suddenly, a tasty clump of eucalyptus falls to the ground, catching their attention. Biff glances at Rocco, who appears to be 15 m away, then over to the eucalyptus, which appears to be 18 m away. From Biff’s point of view, Rocco and the eucalyptus are separated by an angle of 45°. Rocco’s top running speed is 1.0 m/s, but Biff can run one and a half times as fast. Can Biff beat Rocco to the eucalyptus? State any assumptions you make. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841114 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=841114\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><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%0A%0D%0Alocate%28-2%2C4.8%2Cmatrix%281%2C3%2C%22Birds-eye%22%2C%22view%22%2C%22diagram%22%29%29%2C%0D%0A%0D%0Acircle%281%2C-1%2C0.05%29%2Ccircle%281%2C-1%2C0.07%29%2Ccircle%281%2C-1%2C0.09%29%2Ccircle%281%2C-1%2C0.11%29%2Ccircle%281%2C-1%2C0.13%29%2Ccircle%281%2C-1%2C0.15%29%2Ccircle%281%2C-1%2C0.17%29%2Ccircle%281%2C-1%2C0.19%29%2Ccircle%280%2C3%2C0.05%29%2Ccircle%280%2C3%2C0.07%29%2Ccircle%280%2C3%2C0.09%29%2Ccircle%280%2C3%2C0.11%29%2Ccircle%280%2C3%2C0.13%29%2Ccircle%280%2C3%2C0.15%29%2Ccircle%280%2C3%2C0.17%29%2Ccircle%280%2C3%2C0.19%29%2Ccircle%28-2%2C-1%2C0.05%29%2Ccircle%28-2%2C-1%2C0.07%29%2Ccircle%28-2%2C-1%2C0.09%29%2Ccircle%28-2%2C-1%2C0.11%29%2Ccircle%28-2%2C-1%2C0.13%29%2Ccircle%28-2%2C-1%2C0.15%29%2Ccircle%28-2%2C-1%2C0.17%29%2Ccircle%28-2%2C-1%2C0.19%29%2C%0D%0A%0D%0Aline%281%2C-1%2C0%2C3%29%2Cline%280%2C3%2C-2%2C-1%29%2Cline%28-2%2C-1%2C1%2C-1%29%2C%0D%0A%0D%0Alocate%28-2.4%2C-1.26%2C%22B%22%29%2Clocate%280.06%2C3.6%2C%22E%22%29%2Clocate%281.28%2C-1.32%2C%22R%22%29%2Clocate%28-0.58%2C-1.72%2C%2215%22%29%2Clocate%28-1.36%2C-0.22%2C45%5Eo%29%2Clocate%28-1.8%2C1.5%2C%2218%22%29%2C%0D%0A%0D%0Alocate%28-4%2C-4%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "B = Biff's starting location<BR>\n" );
document.write( "R = Rocco's starting location<BR>\n" );
document.write( "E = eucalyptus location\r<BR>\n" );
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document.write( "Distances or side lengths:<BR>\n" );
document.write( "EB = 18 meters<BR>\n" );
document.write( "BR = 15 meters\r<BR>\n" );
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document.write( "Given angle<BR>\n" );
document.write( "angle EBR = 45 degrees\r<BR>\n" );
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document.write( "Let's add a new point A.<BR>\n" );
document.write( "This will be placed on segment BR such that segment EA is perpendicular to segment BR.<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%0A%0D%0Alocate%28-2%2C4.8%2Cmatrix%281%2C3%2C%22Birds-eye%22%2C%22view%22%2C%22diagram%22%29%29%2C%0D%0A%0D%0Acircle%281%2C-1%2C0.05%29%2Ccircle%281%2C-1%2C0.07%29%2Ccircle%281%2C-1%2C0.09%29%2Ccircle%281%2C-1%2C0.11%29%2Ccircle%281%2C-1%2C0.13%29%2Ccircle%281%2C-1%2C0.15%29%2Ccircle%281%2C-1%2C0.17%29%2Ccircle%281%2C-1%2C0.19%29%2Ccircle%280%2C3%2C0.05%29%2Ccircle%280%2C3%2C0.07%29%2Ccircle%280%2C3%2C0.09%29%2Ccircle%280%2C3%2C0.11%29%2Ccircle%280%2C3%2C0.13%29%2Ccircle%280%2C3%2C0.15%29%2Ccircle%280%2C3%2C0.17%29%2Ccircle%280%2C3%2C0.19%29%2Ccircle%28-2%2C-1%2C0.05%29%2Ccircle%28-2%2C-1%2C0.07%29%2Ccircle%28-2%2C-1%2C0.09%29%2Ccircle%28-2%2C-1%2C0.11%29%2Ccircle%28-2%2C-1%2C0.13%29%2Ccircle%28-2%2C-1%2C0.15%29%2Ccircle%28-2%2C-1%2C0.17%29%2Ccircle%28-2%2C-1%2C0.19%29%2C%0D%0A%0D%0Acircle%280%2C-1%2C0.05%29%2Ccircle%280%2C-1%2C0.07%29%2Ccircle%280%2C-1%2C0.09%29%2Ccircle%280%2C-1%2C0.11%29%2Ccircle%280%2C-1%2C0.13%29%2Ccircle%280%2C-1%2C0.15%29%2Ccircle%280%2C-1%2C0.17%29%2Ccircle%280%2C-1%2C0.19%29%2C%0D%0A%0D%0Aline%281%2C-1%2C0%2C3%29%2Cline%280%2C3%2C-2%2C-1%29%2Cline%28-2%2C-1%2C1%2C-1%29%2C%0D%0A%0D%0Aline%280%2C-1%2C0%2C3%29%2C%0D%0A%0D%0Aline%280-0.4%2C-1%2C0-0.4%2C-1%2B0.4%29%2Cline%280-0.4%2C-1%2B0.4%2C0%2C-1%2B0.4%29%2C%0D%0A%0D%0Alocate%28-2.4%2C-1.26%2C%22B%22%29%2Clocate%280.06%2C3.6%2C%22E%22%29%2Clocate%281.28%2C-1.32%2C%22R%22%29%2Clocate%28-0.58%2C-1.72%2C%2215%22%29%2Clocate%28-1.36%2C-0.22%2C45%5Eo%29%2Clocate%28-1.8%2C1.5%2C%2218%22%29%2C%0D%0A%0D%0Alocate%28-0.12%2C-1.2%2C%22A%22%29%2C%0D%0A%0D%0Alocate%28-4%2C-4%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Point A splits triangle REB into two right triangles EAB and EAR.\r<BR>\n" );
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document.write( "Angle ABE = 45 degrees, which leads to angle AEB = 45 degrees as well.\r<BR>\n" );
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document.write( "Use the 45-45-90 triangle template, or the trig ratio cosine, to determine that segment AB = EB/sqrt(2) = 18/sqrt(2) = 12.7279 approximately. <BR>\n" );
document.write( "The segment EA will have the same length because triangle EAB is isosceles.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "AB + AR = BR<BR>\n" );
document.write( "12.7279 + AR = 15<BR>\n" );
document.write( "AR = 15 - 12.7279<BR>\n" );
document.write( "AR = 2.2721 approximately\r<BR>\n" );
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document.write( "Focus your attention on triangle EAR.<BR>\n" );
document.write( "It is a right triangle with these approximate leg lengths<BR>\n" );
document.write( "EA = 12.7279<BR>\n" );
document.write( "AR = 2.2721\r<BR>\n" );
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document.write( "Use the pythagorean theorem to find the hypotenuse ER.<BR>\n" );
document.write( "(EA)^2 + (AR)^2 = (ER)^2<BR>\n" );
document.write( "ER = sqrt( (EA)^2 + (AR)^2 )<BR>\n" );
document.write( "ER = sqrt( (12.7279)^2 + (2.2721)^2 )<BR>\n" );
document.write( "ER = 12.9291<BR>\n" );
document.write( "Like the other decimal values, this is approximate.\r<BR>\n" );
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document.write( "Another way to find segment ER is to use the Law of Cosines on triangle REB.\r<BR>\n" );
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document.write( "-----------------------------------------------------------\r<BR>\n" );
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document.write( "We have these key distances or segment lengths<BR>\n" );
document.write( "EB = 18<BR>\n" );
document.write( "ER = 12.9291 (approx)<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%0A%0D%0Alocate%28-2%2C4.8%2Cmatrix%281%2C3%2C%22Birds-eye%22%2C%22view%22%2C%22diagram%22%29%29%2C%0D%0A%0D%0Acircle%281%2C-1%2C0.05%29%2Ccircle%281%2C-1%2C0.07%29%2Ccircle%281%2C-1%2C0.09%29%2Ccircle%281%2C-1%2C0.11%29%2Ccircle%281%2C-1%2C0.13%29%2Ccircle%281%2C-1%2C0.15%29%2Ccircle%281%2C-1%2C0.17%29%2Ccircle%281%2C-1%2C0.19%29%2Ccircle%280%2C3%2C0.05%29%2Ccircle%280%2C3%2C0.07%29%2Ccircle%280%2C3%2C0.09%29%2Ccircle%280%2C3%2C0.11%29%2Ccircle%280%2C3%2C0.13%29%2Ccircle%280%2C3%2C0.15%29%2Ccircle%280%2C3%2C0.17%29%2Ccircle%280%2C3%2C0.19%29%2Ccircle%28-2%2C-1%2C0.05%29%2Ccircle%28-2%2C-1%2C0.07%29%2Ccircle%28-2%2C-1%2C0.09%29%2Ccircle%28-2%2C-1%2C0.11%29%2Ccircle%28-2%2C-1%2C0.13%29%2Ccircle%28-2%2C-1%2C0.15%29%2Ccircle%28-2%2C-1%2C0.17%29%2Ccircle%28-2%2C-1%2C0.19%29%2C%0D%0A%0D%0Aline%281%2C-1%2C0%2C3%29%2Cline%280%2C3%2C-2%2C-1%29%2Cline%28-2%2C-1%2C1%2C-1%29%2C%0D%0A%0D%0Alocate%28-2.4%2C-1.26%2C%22B%22%29%2Clocate%280.06%2C3.6%2C%22E%22%29%2Clocate%281.28%2C-1.32%2C%22R%22%29%2Clocate%28-0.58%2C-1.72%2C%2215%22%29%2Clocate%28-1.36%2C-0.22%2C45%5Eo%29%2Clocate%28-1.8%2C1.5%2C%2218%22%29%2C%0D%0A%0D%0Alocate%280.68%2C1.86%2C%2212.9291%28approx%29%22%29%2C%0D%0A%0D%0Alocate%28-4%2C-4%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Biff needs to travel 18 meters, and his speed is 1.5*1 = 1.5 m/s since it is 1.5 times faster compared to Rocco's speed (of 1 m/s).\r<BR>\n" );
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document.write( "Biff's travel time is:<BR>\n" );
document.write( "time = distance/rate<BR>\n" );
document.write( "time = EB/1.5<BR>\n" );
document.write( "time = 18/1.5<BR>\n" );
document.write( "time = 12 seconds\r<BR>\n" );
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document.write( "Rocco needs to travel along segment ER = 12.9291, and his speed is 1 m/s, so,<BR>\n" );
document.write( "time = distance/rate<BR>\n" );
document.write( "time = ER/1<BR>\n" );
document.write( "time = 12.9291/1<BR>\n" );
document.write( "time = 12.9291\r<BR>\n" );
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document.write( "-----------------------------------------------------------\r<BR>\n" );
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document.write( "We have these time values<BR>\n" );
document.write( "Biff = 12 seconds<BR>\n" );
document.write( "Rocco = 12.9291 seconds approximately\r<BR>\n" );
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document.write( "Because these time values are so close together, it's practically a tie.\r<BR>\n" );
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document.write( "If we could only select one bear, then the winner is <u>&nbsp;&nbsp;&nbsp;<font color=red size=4>Biff</font>&nbsp;&nbsp;&nbsp;</u>\r<BR>\n" );
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document.write( "-----------------------------------------------------------\r<BR>\n" );
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document.write( "The key assumption made is that each koala bear travels at their top speed right from the start. <BR>\n" );
document.write( "This is of course not realistic. <BR>\n" );
document.write( "The bears need time to build up to their top speed. <BR>\n" );
document.write( "There's no guarantee they maintain the top speed even after reaching it.<BR>\n" );
document.write( "The koalas also need to slow down when arriving at the eucalyptus, or else they would overshoot the goal.\r<BR>\n" );
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document.write( "For a more realistic way to solve this, we'd need to use a kinematic equation. However, such a topic is usually dealt with in physics classrooms.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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