document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=479248>Question 479248</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a door is parabolic in shape and its vertex is 8 feet above the floor. The width of the door way at floor level is 10 feet accordingly, would a 6 foot tall person hit their head, if the person walk through the doorway 2 feet from 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 #328399 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=328399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Set your parabola so that the vertex is at (0,8).  That means the floor is the x-axis and the two intercepts are (-5,0) and (5,0).\r<BR>\n" );
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document.write( "Let the function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f\"> that describes this parabola be defined as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
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document.write( "Substituting the coordinate values from the three points we know must be on the parabola:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(0^2)\ +\ b(0)\ +\ c\ =\ 8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(-5^2)\ +\ b(-5)\ +\ c\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(5^2)\ +\ b(5)\ +\ c\ =\ 0\">\r<BR>\n" );
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document.write( "Simplifying the first equation gives us:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c\ =\ 8\">\r<BR>\n" );
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document.write( "Substituting and simplifying the other two equations results in the following 2X2 linear system:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 25a\ -\ 5b\ =\ -8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 25a\ +\ 5b\ =\ -8\">\r<BR>\n" );
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document.write( "The solution of the system and therefore the coordinates of the parabolic function is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ -0.32\ \ \">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 0\ \ \">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\ =\ 8\">, verification of which I leave as an exercise for the student.\r<BR>\n" );
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document.write( "Therefore the desired function is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ -0.32x^2\ +\ 8\">\r<BR>\n" );
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document.write( "Now if a six-foot tall person walks through the door so that the centerline of the person's body is 2 feet right or left of the centerline of the doorway, does the person hit their head?\r<BR>\n" );
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document.write( "If the person's head were infinitely thin, then it would be a simple matter of determining the value of the function at either <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \pm{2}\">.  However, in practical terms, given that the average six-foot tall person has significantly greater cranial volume than the average algebra student, we'll use 6 inches as a rough approximation of the width of the head, half of which is 3 inches.  Hence, we need to determine the function value at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \pm{2.25}\">\r<BR>\n" );
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document.write( "Again, leaving the verification arithmetic to the student, a 6-foot tall person with a head the shape of a rectangular prism with a width along an axis parallel to the plane of the door of 6 inches would clear the door 2 feet either side of center by approximately 4.5 inches.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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