document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1075057>Question 1075057</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cylindrical biscuit tin has a close-fitting lid which overlaps the tin by 1cm. The radii of the tin and the lid are both x cm. The tin and the lid are made from a thin sheet of metal of area 80&#960; square cm and there is no wastage. The volume of the tin is V cubic cm. Show that V=&#960; (40x-x^2-x^3). Use differentiation to find the positive value of x for which V is stationary.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689753 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=689753\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >= height of the tin in cm.<BR>\n" );
document.write( "The surface area of the tin and lid (in square cm) will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2Ax%5E2%2B2pi%2Ax%2A%28h%2B1%29=80pi ALIGN=MIDDLE  ALT=\"2pi%2Ax%5E2%2B2pi%2Ax%2A%28h%2B1%29=80pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing everything by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28h%2B1%29x=40 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28h%2B1%29x=40\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2B1=%2840-x%5E2%29%2Fx ALIGN=MIDDLE  ALT=\"h%2B1=%2840-x%5E2%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2B1=40%2Fx-x%5E2%2Fx ALIGN=MIDDLE  ALT=\"h%2B1=40%2Fx-x%5E2%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2B1=40%2Fx-x ALIGN=MIDDLE  ALT=\"h%2B1=40%2Fx-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=40%2Fx-1-x ALIGN=MIDDLE  ALT=\"h=40%2Fx-1-x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The volume of the tin as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=pi%2Ax%5E2%2Ah ALIGN=MIDDLE  ALT=\"V%28x%29=pi%2Ax%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the expression previously found for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=pi%2Ax%5E2%2A%2840%2Fx-1-x%29 ALIGN=MIDDLE  ALT=\"V%28x%29=pi%2Ax%5E2%2A%2840%2Fx-1-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28x%29=pi%2840x-x%5E2-x%5E3%29 ALIGN=MIDDLE  ALT=\"V%28x%29=pi%2840x-x%5E2-x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Obviously we want to make the shape with the largest possible volume,<BR>\n" );
document.write( "so we find the radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that will give us maximum volume.<BR>\n" );
document.write( "For that we calculate the derivative:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dV%2Fdx=pi%2840-2x-3x%5E2%29 ALIGN=MIDDLE  ALT=\"dV%2Fdx=pi%2840-2x-3x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The quadratic polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40-2x-3x%5E2=%2810-3x%29%28x%2B4%29 ALIGN=MIDDLE  ALT=\"40-2x-3x%5E2=%2810-3x%29%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10%2F3 ALIGN=MIDDLE  ALT=\"x=10%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "changing from positive to negative at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=10%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=10%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > value that will yield the maximum volume,<BR>\n" );
document.write( "so the radius of the tin should be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2810+%2F+3%29 ALIGN=MIDDLE  ALT=\"highlight%2810+%2F+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283.3cm%29 ALIGN=MIDDLE  ALT=\"highlight%283.3cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: The other zero of the derivative, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is a minimum of the function,<BR>\n" );
document.write( "but a negative value for the radius has no practical meaning.<BR>\n" );
document.write( "The problem tells you to look for the positive value,<BR>\n" );
document.write( "just in case you do not understand the problem, but can follow problem-solving recipes. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );