document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185775>Question 1185775</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Yohan and Jesx want to test the acoustics of a whispering gallery. Yohan is <BR>\n" );
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document.write( "a. What is the highest point of the ceiling from the ground?<BR>\n" );
document.write( "b. How high is the ceiling from the point where Yohan and Jesx stand?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849614 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=849614\">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( "**Understanding the Geometry**\r<BR>\n" );
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document.write( "A whispering gallery is shaped like an ellipse. The foci are the points where whispers can be heard clearly.  The key property we'll use is that the *sum* of the distances from any point on the ellipse to the two foci is constant.\r<BR>\n" );
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document.write( "**a. Finding the Highest Point of the Ceiling**\r<BR>\n" );
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document.write( "1.  **Distance to Foci:** Yohan is 5 feet from the nearest wall (which is also a vertex of the ellipse), and Jesx is 9 feet away from Yohan. This means the distance between the foci (2c) is 9 feet.  Therefore, c = 9/2 = 4.5 feet.\r<BR>\n" );
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document.write( "2.  **Distance from Focus to Vertex:** Yohan is 5 feet from the nearest wall, which is a vertex. The distance from a focus to the nearest vertex is *a - c*, where 'a' is the semi-major axis of the ellipse. So, a - c = 5.\r<BR>\n" );
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document.write( "3.  **Solving for 'a':**  a - 4.5 = 5  => a = 9.5 feet.\r<BR>\n" );
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document.write( "4.  **Center of the Ellipse:** The center of the ellipse is halfway between the foci. Since Yohan is 5 feet from the nearest wall (vertex), and the distance between the foci is 9 feet, the distance from Yohan to the center is (9/2) + 5 - 9.5 = 5 feet.\r<BR>\n" );
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document.write( "5.  **Semi-minor Axis (b):** We can find 'b' using the relationship in an ellipse:  c² = a² - b².<BR>\n" );
document.write( "    4.5² = 9.5² - b²<BR>\n" );
document.write( "    b² = 9.5² - 4.5²<BR>\n" );
document.write( "    b² = 90 - 20.25<BR>\n" );
document.write( "    b²= 70.75<BR>\n" );
document.write( "    b = √70.75 ≈ 8.41 feet\r<BR>\n" );
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document.write( "6.  **Height of the Ceiling:** The highest point of the ceiling is the distance from the center of the ellipse to the top, which is equal to the semi-minor axis *b*. Therefore, the highest point of the ceiling from the ground is approximately 8.41 feet.\r<BR>\n" );
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document.write( "**b. Height of the Ceiling Where Yohan and Jesx Stand**\r<BR>\n" );
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document.write( "1.  **Horizontal Distance:** Yohan is at one focus. The foci are on the horizontal axis.  So, the horizontal distance from the center of the ellipse to where Yohan stands is *c* = 4.5 feet.\r<BR>\n" );
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document.write( "2.  **Equation of the Ellipse:** If we place the center of the ellipse at the origin (0,0), the equation of the ellipse is:\r<BR>\n" );
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document.write( "    (x²/a²) + (y²/b²) = 1<BR>\n" );
document.write( "    (x²/9.5²) + (y²/70.75) = 1\r<BR>\n" );
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document.write( "3.  **Substitute x = 4.5 and solve for y:**  We want to find the height *y* when *x* = 4.5 (where Yohan stands).<BR>\n" );
document.write( "    (4.5²/9.5²) + (y²/70.75) = 1<BR>\n" );
document.write( "    0.225 + (y²/70.75) = 1<BR>\n" );
document.write( "    y²/70.75 = 0.775<BR>\n" );
document.write( "    y² = 54.84<BR>\n" );
document.write( "    y = √54.84 ≈ 7.4 feet\r<BR>\n" );
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document.write( "Therefore, the height of the ceiling where Yohan and Jesx stand is approximately 7.4 feet.\r<BR>\n" );
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document.write( "**Final Answers:**\r<BR>\n" );
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document.write( "*   **a.** The highest point of the ceiling from the ground is approximately 8.41 feet.<BR>\n" );
document.write( "*   **b.** The height of the ceiling from the point where Yohan and Jesx stand is approximately 7.4 feet.<BR>\n" );
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