document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175022>Question 1175022</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A flashlight shape like a paraboloid has its light source at the focus located 1.5cm from the base and is 10cm wide at its opening. How deep is the flashlight at its center? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850627 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=850627\">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. Understand the Paraboloid**\r<BR>\n" );
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document.write( "* A paraboloid is a 3D shape formed by rotating a parabola around its axis of symmetry.<BR>\n" );
document.write( "* The focus is a point inside the paraboloid where all reflected light rays converge.<BR>\n" );
document.write( "* The distance from the vertex (the bottom point) of the paraboloid to the focus is the focal length.\r<BR>\n" );
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document.write( "**2. Set Up the Parabola**\r<BR>\n" );
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document.write( "* We can represent the cross-section of the flashlight as a parabola in a coordinate system.<BR>\n" );
document.write( "* Let the vertex of the parabola be at the origin (0, 0).<BR>\n" );
document.write( "* The focus is at (0, 1.5), meaning the focal length (p) is 1.5 cm.<BR>\n" );
document.write( "* The parabola opens upward, so its equation is x² = 4py.\r<BR>\n" );
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document.write( "**3. Use the Given Dimensions**\r<BR>\n" );
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document.write( "* The flashlight is 10 cm wide at the opening, so the radius is 5 cm.<BR>\n" );
document.write( "* This means when x = 5, we need to find the corresponding y-value, which represents the depth.\r<BR>\n" );
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document.write( "**4. Calculate the Depth**\r<BR>\n" );
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document.write( "* Substitute the focal length (p = 1.5) into the equation: x² = 4(1.5)y, which simplifies to x² = 6y.<BR>\n" );
document.write( "* Substitute x = 5 into this equation: 5² = 6y, so 25 = 6y.<BR>\n" );
document.write( "* Solve for y: y = 25 / 6 ≈ 4.1667 cm.\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The flashlight is approximately 4.17 cm deep at its center.<BR>\n" );
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