document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036306>Question 1036306</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the dimensions that minimize the surface area for a cone with a volume of 225 cm^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650964 by </B><A HREF=/tutors/aboutme.mpl?userid=addingup><B>addingup(3677)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=addingup><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=addingup><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=650964\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Area of the cylinder assuming it has a top and of course it has a bottom:<BR>\n" );
document.write( "A=(2Pi*r*)h + 2(Pi*r^2)  <BR>\n" );
document.write( "= 2Pi*r*h+2Pi*r^2<BR>\n" );
document.write( "Volume of the cylinder:<BR>\n" );
document.write( "V=(Pi*r^2)h<BR>\n" );
document.write( "=Pi*r^2h<BR>\n" );
document.write( "Optimization equation using the formula for the area, since the problem says to minimize the surface area:<BR>\n" );
document.write( "Constraint: Must hold a volume of 225cm^3:<BR>\n" );
document.write( "225=Pi*r^2*h<BR>\n" );
document.write( "h=225/Pi*r^2<BR>\n" );
document.write( "A=2Pi*r(225/Pi*r^2)+2Pi*r^2 <BR>\n" );
document.write( "IN the first equation, Pi in the numerator cancels Pi in the denominator:<BR>\n" );
document.write( "A= 2r(225/r^2)And <BR>\n" );
document.write( "Again in the first equation, r in the top cancels the square in r^2 in the bottom:<BR>\n" );
document.write( "A= 2(225/r)+2Pi*r^2<BR>\n" );
document.write( "A=450/r+2Pi*r^2<BR>\n" );
document.write( "So, we've simplified the equation as much as possible. Let's take the derivative next:<BR>\n" );
document.write( "A=450/r+2Pi*r^2 In the first equation, move the r to the top:<BR>\n" );
document.write( "A=450*r^-1+2Pi*r^2<BR>\n" );
document.write( "A(prime)=-450*r^-2+4Pi*r<BR>\n" );
document.write( "=4Pi*r-450/r^2 Solve for r.<BR>\n" );
document.write( "First, find a common denominator. Multiply both sides times r^2. You get:<BR>\n" );
document.write( "0=(4Pi*r^3-450)/r^2<BR>\n" );
document.write( "&#8706;=4(Pi*r^3-112.5)Divide both sides by 4, then add 112.5 on both sides:<BR>\n" );
document.write( "112.5=Pi*r^3<BR>\n" );
document.write( "112.5/Pi=r^3<BR>\n" );
document.write( "r=cuberoot500/Pi This is the radius that minimizes the surface of the can.<BR>\n" );
document.write( "h=225/Pi(cuberoot112.5/Pi)^2<BR>\n" );
document.write( "=225/Pi(112.5/Pi)^2/3 This is the height of the can that minimizes the surface.<BR>\n" );
document.write( "Run the numbers in your calculator.\r<BR>\n" );
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