document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196361>Question 1196361</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Opposite Flying Plane Problem: Two planes flying opposite directions (North and South) pass each other 80 miles apart at the same altitude. The Northbound plane is flying 200 mph and the Southbound plane is flying 150 mph. a) How far apart are the planes in 20 minutes? b) When are the planes 300 miles apart?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829189 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=829189\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "Let's label the planes as PlaneA and PlaneB\r<BR>\n" );
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document.write( "PlaneA = northbound at 200 mph<BR>\n" );
document.write( "PlaneB = southbound at 150 mph\r<BR>\n" );
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document.write( "Place planeA at (0,0) on the xy grid.<BR>\n" );
document.write( "PlaneB will go to (80,0) which is 80 units away, to represent the 80 mile gap.\r<BR>\n" );
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document.write( "t = number of hours that elapse, after the planes are side by side\r<BR>\n" );
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document.write( "The location of planeA is (0,200t) since it travels 200 mph and it's heading north, i.e. along the positive y axis.<BR>\n" );
document.write( "I'm using the idea that distance = rate*time<BR>\n" );
document.write( "For instance, after t = 1 hour, plane A is at location (0,200t) = (0,200*1) = (0,200)\r<BR>\n" );
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document.write( "Meanwhile, the location of planeB is (80,-150t)<BR>\n" );
document.write( "The negative is to indicate going south\r<BR>\n" );
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document.write( "20 min = 20/60 = 1/3 of an hour\r<BR>\n" );
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document.write( "Let's plug in t = 1/3 to find the location of each plane<BR>\n" );
document.write( "A = (0,200t) = (0,200*(1/3)) = (0,200/3)<BR>\n" );
document.write( "B = (80,-150t) = (80,-150*(1/3)) = (80,-50)\r<BR>\n" );
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document.write( "Now compute the distance from A to B<BR>\n" );
document.write( "d = sqrt( (x1-x2)^2 + (y1-y2)^2 )<BR>\n" );
document.write( "d = sqrt( (0-80)^2 + (200/3-(-50))^2 )<BR>\n" );
document.write( "d = sqrt( (0-80)^2 + (200/3+50)^2 )<BR>\n" );
document.write( "d = sqrt( 20,011.1111111111 )<BR>\n" );
document.write( "d = 141.46063449282<BR>\n" );
document.write( "d = 141.4606\r<BR>\n" );
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document.write( "Answer: Approximately 141.4606 miles<BR>\n" );
document.write( "Round this however your teacher instructs.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Let's go back to <BR>\n" );
document.write( "A = (0,200t)<BR>\n" );
document.write( "B = (80,-150t)<BR>\n" );
document.write( "and compute the distance based on these coordinates\r<BR>\n" );
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document.write( "d = sqrt( (x1-x2)^2 + (y1-y2)^2 )<BR>\n" );
document.write( "d = sqrt( (0-80)^2 + (200t-(-150t))^2 )<BR>\n" );
document.write( "d = sqrt( 6400 + (200t+150t)^2 )<BR>\n" );
document.write( "d = sqrt( 6400 + (350t)^2 )<BR>\n" );
document.write( "d = sqrt( 6400 + 122,500t^2 )\r<BR>\n" );
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document.write( "As a slight detour, let's plug in t = 1/3 to get...<BR>\n" );
document.write( "d = sqrt( 6400 + 122,500t^2 )<BR>\n" );
document.write( "d = sqrt( 6400 + 122,500(1/3)^2 )<BR>\n" );
document.write( "d = 141.46063449282<BR>\n" );
document.write( "We get the same result as before, so it helps confirm we have the correct general distance equation.\r<BR>\n" );
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document.write( "What we'll do from here is plug in d = 300 and solve for t<BR>\n" );
document.write( "d = sqrt( 6400 + 122,500t^2 )<BR>\n" );
document.write( "300 = sqrt( 6400 + 122,500t^2 )<BR>\n" );
document.write( "300^2 = 6400 + 122,500t^2<BR>\n" );
document.write( "90,000 = 6400 + 122,500t^2<BR>\n" );
document.write( "6400 + 122,500t^2 = 90,000<BR>\n" );
document.write( "122,500t^2 = 90,000<BR>\n" );
document.write( "122,500t^2 = 90,000 - 6400<BR>\n" );
document.write( "122,500t^2 = 83,600<BR>\n" );
document.write( "t^2 = (83,600)/(122,500)<BR>\n" );
document.write( "t^2 = 0.68244897959183<BR>\n" );
document.write( "t = sqrt(0.68244897959183)<BR>\n" );
document.write( "t = 0.82610470256006\r<BR>\n" );
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document.write( "The planes are 300 miles apart at the time marker of approximately 0.82610470256006 hours\r<BR>\n" );
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document.write( "Multiply by 60 to convert to minutes<BR>\n" );
document.write( "60*0.82610470256006 = 49.5662821536037\r<BR>\n" );
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document.write( "Answer: Approximately 49.5663 minutes<BR>\n" );
document.write( "Round this value however your teacher instructs.<BR>\n" );
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