document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823745>Question 823745</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangle has one vertex in quadrant I on the graph of y=10-x^2, another at the origin, one on the positive x-axis, and one on the positive y-axis.  <BR>\n" );
document.write( "a.) Express the area A of the rectangle as a function of x.<BR>\n" );
document.write( "b.) Find the largest area A that can be enclosed by the rectangle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495897 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=495897\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The parabola looks like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C450%2C-5%2C5%2C-2%2C13%2C10-x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C450%2C-5%2C5%2C-2%2C13%2C10-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with x-intercepts at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2810%29 ALIGN=MIDDLE  ALT=\"sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2810%29 ALIGN=MIDDLE  ALT=\"-sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2810%29 ALIGN=MIDDLE  ALT=\"sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = approx. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.16 ALIGN=MIDDLE  ALT=\"3.16\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "The rectangle in the first quadrant looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C5%2C-2%2C8%2C%0D%0Agrid%280%29%2C%0D%0Ared%28circle%281.83%2C6.67%2C0.1%29%29%2C%0D%0Alocate%281.9%2C7%2CP%28x%2Cy%29%29%2C%0D%0Arectangle%280%2C0%2C1.83%2C20%2F3%29%2C%0D%0Alocate%280.85%2C6.7%2Cx%29%2C%0D%0Alocate%281.9%2C3.5%2Cy=10-x%5E2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > We want a formula to calculate the area of that rectangle, and we want to find the maximum area.<BR>\n" );
document.write( " <BR>\n" );
document.write( "a.) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=x%2Ay=x%2810-x%5E2%29=10x-x%5E3 ALIGN=MIDDLE  ALT=\"A%28x%29=x%2Ay=x%2810-x%5E2%29=10x-x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3Csqrt%2810%29 ALIGN=MIDDLE  ALT=\"0%3Cx%3Csqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We have to put that restriction on the domain, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%2Cy%29 ALIGN=MIDDLE  ALT=\"P%28x%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be in the first quadrant.<BR>\n" );
document.write( "So we can write the function as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A%28x%29=-x%5E3%2B10x%29 ALIGN=MIDDLE  ALT=\"highlight%28A%28x%29=-x%5E3%2B10x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280%3Cx%3Csqrt%2810%29%29 ALIGN=MIDDLE  ALT=\"highlight%280%3Cx%3Csqrt%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is a polynomial function.<BR>\n" );
document.write( "Fully factored, it can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=-x%28x%2Bsqrt%2810%29%29%28x-sqrt%2810%29%29 ALIGN=MIDDLE  ALT=\"A%28x%29=-x%28x%2Bsqrt%2810%29%29%28x-sqrt%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the domain were not restricted, we would say that it has zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%2810%29%7D%7D%2C+%7B%7B%7Bx=0 ALIGN=MIDDLE  ALT=\"x=-sqrt%2810%29%7D%7D%2C+%7B%7B%7Bx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%2810%29 ALIGN=MIDDLE  ALT=\"x=sqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We would figure out that it is positive and decreasing for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-sqrt%2810%29%7D%7D%2C%0D%0Awhere+%7B%7B%7Bx%3C0 ALIGN=MIDDLE  ALT=\"x%3C-sqrt%2810%29%7D%7D%2C%0D%0Awhere+%7B%7B%7Bx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bsqrt%2810%29%3C0 ALIGN=MIDDLE  ALT=\"x%2Bsqrt%2810%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bsqrt%2810%29%3C0 ALIGN=MIDDLE  ALT=\"x%2Bsqrt%2810%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is negative for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%2810%29%3Cx%3C0 ALIGN=MIDDLE  ALT=\"-sqrt%2810%29%3Cx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and going through a minimum in that interval.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3Csqrt%2810%29 ALIGN=MIDDLE  ALT=\"0%3Cx%3Csqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29+is+positive+and+goes+through+a+maximum.%0D%0AFinally%2C+for+%7B%7B%7Bx%3Esqrt%2810%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , the function is negative and decreasing.<BR>\n" );
document.write( "Here's the graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C300%2C-5%2C5%2C-15%2C15%2C-x%5E3%2B10x%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C300%2C-5%2C5%2C-15%2C15%2C-x%5E3%2B10x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "andpositive for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3Esqrt%2810%29 ALIGN=MIDDLE  ALT=\"x%3Esqrt%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "b.) The only way that I know to find the maximum area of such a rectangle requires using calculus and finding the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29 ALIGN=MIDDLE  ALT=\"A%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dA%2Fdt=-3x%5E2%2B10 ALIGN=MIDDLE  ALT=\"dA%2Fdt=-3x%5E2%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The zeros of that derivative show us the location of the minimum and maximum.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B10=0 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B10=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10=3x%5E2 ALIGN=MIDDLE  ALT=\"10=3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F3=x%5E2 ALIGN=MIDDLE  ALT=\"10%2F3=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The solutions are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%2810%2F3%29=-sqrt%2830%29%2F3 ALIGN=MIDDLE  ALT=\"x=-sqrt%2810%2F3%29=-sqrt%2830%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%2810%2F3%29=sqrt%2830%29%2F3 ALIGN=MIDDLE  ALT=\"x=sqrt%2810%2F3%29=sqrt%2830%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The maximum occurs at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%2830%29%2F3 ALIGN=MIDDLE  ALT=\"x=sqrt%2830%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=10%2F3 ALIGN=MIDDLE  ALT=\"x%5E2=10%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting those values into <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=x%2810-x%5E2%29 ALIGN=MIDDLE  ALT=\"A%28x%29=x%2810-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can easily calculate<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bmax%5D=%28sqrt%2830%29%2F3%29%2810-10%2F3%29=%28sqrt%2830%29%2F3%29%2820%2F3%29=highlight%2820sqrt%2830%29%2F9%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate value is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2812.17%29 ALIGN=MIDDLE  ALT=\"highlight%2812.17%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );