document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190079>Question 1190079</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem. \r<BR>\n" );
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document.write( "The dome of a whispering gallery has the form of a semi-ellipse so that the two person standing at the foci will be able to hear each other. This is because sound waves from one focus, when they reach the semi-elliptical ceiling, bounce off to the other focus. If one such whispering gallery has a focus of 3m away from the end of the semi-ellipse, and the foci are 14m away from each other, how high is the dome at the center? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849203 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=849203\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "1. **Visualize:** Imagine the semi-ellipse as a horizontal slice of a full ellipse. The two people are standing at the foci (F1 and F2). The distance from a focus to the nearest end of the semi-ellipse is given as 3m. The distance between the foci (2c) is 14m.\r<BR>\n" );
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document.write( "2. **Find 'c':** The distance from the center of the ellipse to each focus is 'c'. Since the foci are 14m apart, 2c = 14m, so c = 7m.\r<BR>\n" );
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document.write( "3. **Find 'a':** The distance from the center to the end of the semi-ellipse along the major axis is 'a'. We know that the distance from a focus to the nearest end is 3m. Since the distance from the center to the focus is 7m, we have:\r<BR>\n" );
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document.write( "   a = c + 3m = 7m + 3m = 10m\r<BR>\n" );
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document.write( "4. **Find 'b':**  'b' is the distance from the center to the highest point of the semi-ellipse (the height of the dome at the center). We use the relationship for ellipses:\r<BR>\n" );
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document.write( "   a² = b² + c²\r<BR>\n" );
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document.write( "   10² = b² + 7²\r<BR>\n" );
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document.write( "   100 = b² + 49\r<BR>\n" );
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document.write( "   b² = 100 - 49\r<BR>\n" );
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document.write( "   b² = 51\r<BR>\n" );
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document.write( "   b = √51 ≈ 7.14m\r<BR>\n" );
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document.write( "Therefore, the dome is approximately 7.14 meters high at the center.<BR>\n" );
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