document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170522>Question 1170522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The cable of suspension bridge hangs in the shape of a parabola. The towers supporting the cable are 400ft apart and 150ft high. If the cable, at its lowest, is 30ft above the bridge at its midpoint, how high is the cable 50ft away (horizontally) from either tower? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851158 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Set up a Coordinate System**\r<BR>\n" );
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document.write( "* Place the vertex of the parabola (the lowest point of the cable) at the origin (0, 30).<BR>\n" );
document.write( "* The towers are 400 feet apart, so they are located at (-200, 150) and (200, 150).\r<BR>\n" );
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document.write( "**2. Find the Equation of the Parabola**\r<BR>\n" );
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document.write( "* The general equation of a parabola with a vertical axis of symmetry and vertex (h, k) is:<BR>\n" );
document.write( "   (x - h)^2 = 4p(y - k)<BR>\n" );
document.write( "* In our case, the vertex is (0, 30), so the equation becomes:<BR>\n" );
document.write( "   x^2 = 4p(y - 30)\r<BR>\n" );
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document.write( "* We know the towers are at (200, 150). Plug these coordinates into the equation to find 'p':<BR>\n" );
document.write( "   (200)^2 = 4p(150 - 30)<BR>\n" );
document.write( "   40000 = 4p(120)<BR>\n" );
document.write( "   40000 = 480p<BR>\n" );
document.write( "   p = 40000 / 480 = 83.333... = 250/3\r<BR>\n" );
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document.write( "* Now, plug 'p' back into the equation:<BR>\n" );
document.write( "   x^2 = 4(250/3)(y - 30)<BR>\n" );
document.write( "   x^2 = (1000/3)(y - 30)\r<BR>\n" );
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document.write( "**3. Find the Height 50 Feet from a Tower**\r<BR>\n" );
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document.write( "* We want to find the height of the cable 50 feet horizontally from either tower. Since the towers are at x = ±200, we're looking for the height at x = 200 - 50 = 150 and x = -200 + 50 = -150. Due to symmetry, the height will be the same.<BR>\n" );
document.write( "* Plug x = 150 (or x = -150) into the parabola equation:<BR>\n" );
document.write( "   (150)^2 = (1000/3)(y - 30)<BR>\n" );
document.write( "   22500 = (1000/3)(y - 30)<BR>\n" );
document.write( "   22500 * (3/1000) = y - 30<BR>\n" );
document.write( "   67.5 = y - 30<BR>\n" );
document.write( "   y = 67.5 + 30<BR>\n" );
document.write( "   y = 97.5\r<BR>\n" );
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document.write( "**4. Answer**\r<BR>\n" );
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document.write( "* The cable is 97.5 feet high 50 feet away (horizontally) from either tower.\r<BR>\n" );
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document.write( "**Final Answer:** 97.5 feet<BR>\n" );
document.write( " </SPAN>\n" );
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