document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169029>Question 1169029</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two control towers are located at points Q(-400,0) and R(400,0), on a straight shore <BR>\n" );
document.write( "where the x-axis runs through (all distances are in meters). At the same moment, both <BR>\n" );
document.write( "towers sent a radio signal to a ship out at sea, each traveling at 250 m/µs. The ship <BR>\n" );
document.write( "received the signal from Q 2.9 µs (microseconds) before the message from R. Find the <BR>\n" );
document.write( "equation of the curve containing the possible location of the ship. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #793802 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=793802\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Ok, I'll give this one a try. I was reminded of the definition of the hyperbola<BR>\n" );
document.write( "when the problem mentioned a fixed time difference between the signals arriving<BR>\n" );
document.write( "from Q vs. the signals from R. The hyperbola is the set of all points<BR>\n" );
document.write( "(x,y) such that the difference of the distances from (x,y) to the foci is constant. If we <BR>\n" );
document.write( "place a tower at each focus, then we can satisfy the conditions <BR>\n" );
document.write( "of the problem with a hyperbola to describe the possible locations of the ship.<BR>\n" );
document.write( "The difference in the distances between a point on the curve and the<BR>\n" );
document.write( "two foci = 2a, for the vertex at (a,0). The foci are at the points (-c,0) and (c,0), and b = sqrt(c^2 - a^2).<BR>\n" );
document.write( "Since the signal velocity is 250 m/us, the difference in the travel times <BR>\n" );
document.write( "from each of the towers is <BR>\n" );
document.write( "d2/250 - d1/250 -> d2 - d1 = 250 m/us * 2.9 us = 2a = 725.<BR>\n" );
document.write( "Thus d2 - d1 = 250*2.9 = 725 m. Thus a = 725/2 = 362.5<BR>\n" );
document.write( "Therefore b = sqrt(400^2 - 362.5^2) = 169.09687<BR>\n" );
document.write( "The equation for a hyperbola centered at the origin is (x/a)^2 - (y/b)^2 = 1.<BR>\n" );
document.write( "So the possible locations for the ship are given by:<BR>\n" );
document.write( "(x/362.5)^2 - (y/169.09687)^2 = 1. </SPAN>\n" );
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