document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=562747>Question 562747</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Academic Calculus- Pythagorean Theorem and Related Rates Word Problem:\r<BR>\n" );
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document.write( "A plane takes off from airport A and flies east at 480 mph. At the moment of takeoff, a second plane, approaching A from the north at 320 mph, is still 240 miles from A. Fifteen minutes later is the distance between the planes increasing or decreasing, and how fast? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #364671 by </B><A HREF=/tutors/aboutme.mpl?userid=issacodegard><B>issacodegard(60)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=issacodegard><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/cgi-bin/show-face.mpl?user=issacodegard><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=364671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi, let A=(0,0).  Then, the second plane starts at (0,240).  Let x(t) be the distance from A to plane 1 at time t, y(t) be the distance from A to plane 2 at time 2, and z(t) be the distance between the planes at time t (I say distance because I know that plane two doesn't reach the airport in 15 minutes). We know,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=z%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=z%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dx%29%2F%28dt%29=480 ALIGN=MIDDLE  ALT=\"%28dx%29%2F%28dt%29=480\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%29%2F%28dt%29=-320 ALIGN=MIDDLE  ALT=\"%28dy%29%2F%28dt%29=-320\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So by implicit differentiation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%28%28dx%29%2F%28dt%29%29%2B2y%28%28dy%29%2F%28dt%29%29=2z%28%28dz%29%2F%28dt%29%29 ALIGN=MIDDLE  ALT=\"2x%28%28dx%29%2F%28dt%29%29%2B2y%28%28dy%29%2F%28dt%29%29=2z%28%28dz%29%2F%28dt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So at time t=1/4 hr,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=480%281%2F4%29=120 ALIGN=MIDDLE  ALT=\"x=480%281%2F4%29=120\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=240-320%281%2F4%29=160 ALIGN=MIDDLE  ALT=\"y=240-320%281%2F4%29=160\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=sqrt%28120%5E2%2B160%5E2%29=200 ALIGN=MIDDLE  ALT=\"z=sqrt%28120%5E2%2B160%5E2%29=200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dz%29%2F%28dt%29=%281%2F200%29%28120%2A480%2B160%2A%28-320%29%29=32 ALIGN=MIDDLE  ALT=\"%28dz%29%2F%28dt%29=%281%2F200%29%28120%2A480%2B160%2A%28-320%29%29=32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is positive, so the distance between them is increasing, so the planes are getting further away at a rate of 32mph. </SPAN>\n" );
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