document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168412>Question 1168412</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Bayonne bridge connects Staten Island, New York to New Jersey. It has an arch in the shape of parabola that opens downward. Write an equation of the parabola to model the arch, assuming that the origin is at the surface of the water. 325. ft and 1675ft ​ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851571 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=851571\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem and find the equation of the parabola.\r<BR>\n" );
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document.write( "**1. Understand the Problem**\r<BR>\n" );
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document.write( "* The arch of the Bayonne Bridge is in the shape of a downward-opening parabola.<BR>\n" );
document.write( "* The origin (0, 0) is at the surface of the water.<BR>\n" );
document.write( "* We are given two key dimensions:<BR>\n" );
document.write( "    * The width of the arch at the water's surface is 1675 feet.<BR>\n" );
document.write( "    * The height of the arch is 325 feet.\r<BR>\n" );
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document.write( "**2. Set Up the Vertex**\r<BR>\n" );
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document.write( "* Since the parabola opens downward and the origin is at the water's surface, the vertex of the parabola will be at the highest point of the arch.<BR>\n" );
document.write( "* The vertex will be located at (1675/2, 325) = (837.5, 325).\r<BR>\n" );
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document.write( "**3. General Equation of a Parabola**\r<BR>\n" );
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document.write( "The general equation of a parabola with a vertical axis of symmetry is:\r<BR>\n" );
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document.write( "y = a(x - h)² + k\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* (h, k) is the vertex of the parabola.<BR>\n" );
document.write( "* 'a' determines the direction and width of the parabola.\r<BR>\n" );
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document.write( "**4. Plug in the Vertex**\r<BR>\n" );
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document.write( "* We know the vertex is (837.5, 325), so h = 837.5 and k = 325.<BR>\n" );
document.write( "* The equation becomes:\r<BR>\n" );
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document.write( "y = a(x - 837.5)² + 325\r<BR>\n" );
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document.write( "**5. Find 'a'**\r<BR>\n" );
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document.write( "* We know the parabola passes through the origin (0, 0). Plug in x = 0 and y = 0:\r<BR>\n" );
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document.write( "0 = a(0 - 837.5)² + 325<BR>\n" );
document.write( "0 = a(837.5)² + 325<BR>\n" );
document.write( "-325 = a(837.5)²<BR>\n" );
document.write( "a = -325 / (837.5)²<BR>\n" );
document.write( "a = -325 / 701406.25<BR>\n" );
document.write( "a ≈ -0.00046335\r<BR>\n" );
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document.write( "**6. Write the Equation**\r<BR>\n" );
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document.write( "* Substitute the value of 'a' back into the equation:\r<BR>\n" );
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document.write( "y = -0.00046335(x - 837.5)² + 325\r<BR>\n" );
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document.write( "**7. Simplify (Optional)**\r<BR>\n" );
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document.write( "* If needed, we can round 'a' to a more manageable value:\r<BR>\n" );
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document.write( "y ≈ -0.000463(x - 837.5)² + 325\r<BR>\n" );
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document.write( "**Therefore, the equation of the parabola that models the arch of the Bayonne Bridge is approximately y = -0.000463(x - 837.5)² + 325.**<BR>\n" );
document.write( " </SPAN>\n" );
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