document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206901>Question 1206901</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 288 feet. The smallest diameter of the tower is 143 feet which is 393.5 feet above the ground. The tower is 581 feet tall.\r<BR>\n" );
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document.write( "Find the width of the tower at a height of 38 feet. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847940 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=847940\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem, we need to set up a coordinate system where the origin (0,0) is at the center of the bottom of the tower. The hyperbola will have a vertical transverse axis. \r<BR>\n" );
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document.write( "**Step 1: Determine the equation of the hyperbola.**\r<BR>\n" );
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document.write( "Given the dimensions of the tower, we can determine the values of a and b in the standard equation of a hyperbola with a vertical transverse axis:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "(y^2/a^2) - (x^2/b^2) = 1<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "* **a:** Half the distance between the vertices (top and bottom of the tower) = (581 - 393.5) / 2 = 93.75<BR>\n" );
document.write( "* **b:** Half the diameter of the smallest part of the tower = 143 / 2 = 71.5\r<BR>\n" );
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document.write( "So, the equation of the hyperbola is:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "(y^2/93.75^2) - (x^2/71.5^2) = 1<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Step 2: Find the width at a height of 38 feet.**\r<BR>\n" );
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document.write( "At a height of 38 feet, y = 38. We need to solve for x:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "(38^2/93.75^2) - (x^2/71.5^2) = 1<BR>\n" );
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document.write( "Solving for x, we get:\r<BR>\n" );
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document.write( "x ≈ ± 53.8<BR>\n" );
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document.write( "The width at a height of 38 feet is 2 * 53.8 = **107.6 feet**.<BR>\n" );
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