document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165709>Question 1165709</A>: <I><SPAN STYLE=\"word-break: break-all;\">  arch in an amusement park has a parabolic shape. it has a height of 74 feet and the base width of 38 feet. what is the height of the Arch 8 feet from 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 #852940 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=852940\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "# Given parameters<BR>\n" );
document.write( "H = 74  # Maximum height (feet)<BR>\n" );
document.write( "W = 38  # Base width (feet)<BR>\n" );
document.write( "x = 8   # Horizontal distance from the center (feet)\r<BR>\n" );
document.write( "\n" );
document.write( "# The parabola is defined by y = a*x^2 + H.<BR>\n" );
document.write( "# The base endpoints are at (W/2, 0) and (-W/2, 0).<BR>\n" );
document.write( "# W/2 = 38 / 2 = 19<BR>\n" );
document.write( "# We use the point (19, 0) to find 'a'.<BR>\n" );
document.write( "# 0 = a * (19)^2 + 74<BR>\n" );
document.write( "# a = -74 / 19^2 = -74 / 361\r<BR>\n" );
document.write( "\n" );
document.write( "# The equation of the parabola is: y = (-74/361) * x^2 + 74\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate the height (y) at x = 8<BR>\n" );
document.write( "height = (-74 / 361) * (x**2) + H\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"Height at 8 feet from the center: {height} feet\")\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Height at 8 feet from the center: 60.880886426592795 feet\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "The height of the arch 8 feet from the center is approximately **60.88 feet**.\r<BR>\n" );
document.write( "\n" );
document.write( "Here is the step-by-step calculation:\r<BR>\n" );
document.write( "\n" );
document.write( "### 1\. Establish the Equation of the Parabola\r<BR>\n" );
document.write( "\n" );
document.write( "We model the parabolic arch using a standard coordinate system where the vertex (maximum height) is on the $y$-axis and the base is on the $x$-axis.\r<BR>\n" );
document.write( "\n" );
document.write( "The general equation for a parabola opening downward with a vertex at $(0, h)$ is:<BR>\n" );
document.write( "$$y = ax^2 + h$$\r<BR>\n" );
document.write( "\n" );
document.write( "  * **Height ($h$):** $74$ feet.<BR>\n" );
document.write( "  * **Base Width:** $38$ feet, meaning the arch touches the ground (where $y=0$) at $x = \pm \frac{38}{2} = \pm 19$.\r<BR>\n" );
document.write( "\n" );
document.write( "Using the point $(19, 0)$ and the vertex $h=74$:<BR>\n" );
document.write( "$$0 = a(19)^2 + 74$$<BR>\n" );
document.write( "$$0 = 361a + 74$$<BR>\n" );
document.write( "$$a = -\frac{74}{361}$$\r<BR>\n" );
document.write( "\n" );
document.write( "The equation for the arch is:<BR>\n" );
document.write( "$$y = -\frac{74}{361}x^2 + 74$$\r<BR>\n" );
document.write( "\n" );
document.write( "### 2\. Calculate the Height\r<BR>\n" );
document.write( "\n" );
document.write( "To find the height $y$ at a horizontal distance of $x=8$ feet from the center:\r<BR>\n" );
document.write( "\n" );
document.write( "$$y = -\frac{74}{361}(8)^2 + 74$$<BR>\n" );
document.write( "$$y = -\frac{74}{361}(64) + 74$$<BR>\n" );
document.write( "$$y = 74 - \frac{4736}{361}$$<BR>\n" );
document.write( "$$y \approx 74 - 13.11911$$<BR>\n" );
document.write( "$$y \approx 60.88088 \text{ feet}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );