document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=310688>Question 310688</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello I have a problem solving this parabola question.The path of a cliff diver as he dives into a lake ,is given by this eqaution,y=-(x-10)(SQAURED)+75,where y metres is the diver's height above the water and,x metres is the horisontal distance travelled by the diver.What is the maximum height the diver is above the water? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #222183 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=222183\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Not sure whether you just need an intuitive approach, a formal algebraic approach, or a calculus approach.\r<BR>\n" );
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document.write( "<u>Intuitive</u>\r<BR>\n" );
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document.write( "The largest y can be is 75 and that is when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x\ -\ 10)^2\ =\ 0\">.  That is because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x\ -\ 10)^2\"> is always a positive number, unless it is zero, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -(x\ -\ 10)^2\"> is always a negative number unless it is zero. Therefore you are always subtracting something from 75 unless <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x\ -\ 10)^2\ =\ 0\">, which happens when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 10\">\r<BR>\n" );
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document.write( "While we are at it -- 75 <b><i>meters</i></b>? That is 246 feet.  Hitting the water from that height would be like hitting concrete.  I rather think your cliff diver is only going to perform this particular stunt once.\r<BR>\n" );
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document.write( "<u>Algebraic</u>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -\left(x\ -\ 10\right)^2\ +\ 75\">\r<BR>\n" );
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document.write( "Expand the binomial expression:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -x^2\ +\ 20x\ -\ 100\ + 75\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -x^2\ +\ 20x\ -\ 25\">\r<BR>\n" );
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document.write( "A parabola with a negative lead coefficient opens downward, hence the vertex is a maximum.  The vertex of a parabola expressed in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ + bx\ +\ c\"> form has a vertex that occurs at an <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{-b}{2a}\">, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_v\ =\ \frac{-b}{2a}\ =\ \frac{-20}{-2}\ =\ 10\">\r<BR>\n" );
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document.write( "And the value of the function at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 10\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ f(10)\ =\ -(10)^2\ +\ 20(10)\ -\ 25\ =\ 75\">\r<BR>\n" );
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document.write( "<u>Calculus</u>\r<BR>\n" );
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document.write( "Using the function definition from the Algebraic discussion:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -x^2\ +\ 20x\ -\ 25\">\r<BR>\n" );
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document.write( "The function will have a maximum where the first derivative is zero and the second derivative is negative.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dy}{dx}\ =\ -2x\ +\ 20\">\r<BR>\n" );
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document.write( "Set it equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -2x\ +\ 20\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \frac{d^2y}{dx^2}\ =\ -2\">\r<BR>\n" );
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document.write( "which is negative for all <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> in the domain of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">\r<BR>\n" );
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document.write( "So there is a maximum at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ f(10)\ =\ -(10)^2\ +\ 20(10)\ -\ 25\ =\ 75\">\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" );
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