document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171510>Question 1171510</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show the mathematics of designing and constrcting a parabolic arch that is 10 meters across the base and 10 meters tall (measured from the ground to the vertex at the bottom of the keystone). Remember: -4a(y-k)=(x-h)^2<BR>\n" );
document.write( "a) Place an axis system on this parabola<BR>\n" );
document.write( "b) Find the function for this parabola<BR>\n" );
document.write( "c) Make a chart of values for functional heights spaced 1 meter apart<BR>\n" );
document.write( "d)How far down from the vertex is the focal point? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850900 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=850900\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down the design and construction of the parabolic arch step-by-step.\r<BR>\n" );
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document.write( "**a) Placing an Axis System**\r<BR>\n" );
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document.write( "* We'll place the vertex of the parabola at the origin (0, 0).<BR>\n" );
document.write( "* The x-axis will run along the base of the arch.<BR>\n" );
document.write( "* The y-axis will run vertically through the center of the arch.<BR>\n" );
document.write( "* The base of the arch is 10 meters across, so the endpoints of the base are at (-5, -10) and (5, -10).<BR>\n" );
document.write( "* The arch is 10 meters tall, so the vertex is at (0, 0).\r<BR>\n" );
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document.write( "**b) Finding the Function for the Parabola**\r<BR>\n" );
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document.write( "* We'll use the standard form of a vertical parabola: -4a(y - k) = (x - h)²<BR>\n" );
document.write( "* Vertex (h, k) = (0, 0)<BR>\n" );
document.write( "* The equation becomes: -4ay = x²<BR>\n" );
document.write( "* We know a point on the parabola is (5, -10). Let's plug this in:<BR>\n" );
document.write( "    * -4a(-10) = 5²<BR>\n" );
document.write( "    * 40a = 25<BR>\n" );
document.write( "    * a = 25/40 = 5/8<BR>\n" );
document.write( "* Substitute 'a' back into the equation:<BR>\n" );
document.write( "    * -4(5/8)y = x²<BR>\n" );
document.write( "    * -5/2 y = x²<BR>\n" );
document.write( "    * y = (-2/5)x²\r<BR>\n" );
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document.write( "**c) Chart of Values for Functional Heights (1 Meter Spacing)**\r<BR>\n" );
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document.write( "| x (meters) | y (meters) |<BR>\n" );
document.write( "| :--------- | :--------- |<BR>\n" );
document.write( "| -5         | -10        |<BR>\n" );
document.write( "| -4         | -6.4       |<BR>\n" );
document.write( "| -3         | -3.6       |<BR>\n" );
document.write( "| -2         | -1.6       |<BR>\n" );
document.write( "| -1         | -0.4       |<BR>\n" );
document.write( "| 0          | 0          |<BR>\n" );
document.write( "| 1          | -0.4       |<BR>\n" );
document.write( "| 2          | -1.6       |<BR>\n" );
document.write( "| 3          | -3.6       |<BR>\n" );
document.write( "| 4          | -6.4       |<BR>\n" );
document.write( "| 5          | -10        |\r<BR>\n" );
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document.write( "**d) Focal Point**\r<BR>\n" );
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document.write( "* The distance from the vertex to the focal point is 'a'.<BR>\n" );
document.write( "* We found a = 5/8 meters.<BR>\n" );
document.write( "* Therefore, the focal point is 5/8 meters (0.625 meters) below the vertex.\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **Axis System:** Vertex at (0, 0), base endpoints at (-5, -10) and (5, -10).<BR>\n" );
document.write( "* **Parabola Function:** y = (-2/5)x²<BR>\n" );
document.write( "* **Chart of Values:** (See table above)<BR>\n" );
document.write( "* **Focal Point:** 5/8 meters (0.625 meters) below the vertex.<BR>\n" );
document.write( " </SPAN>\n" );
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