document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011000>Question 1011000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the surface area of the cone z=sqrt(x^2+y^2) below the plane z=8.<BR>\n" );
document.write( "Please show your solution step by step. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #626542 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=626542\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We want the surface area of the portion of the cone z^2 = x^2 + y^2 between z=0 and z=8.  The equation of the cone in cylindrical coordinates is just z = r, so we can take as our parameters r and t (representing theta).<BR>\n" );
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document.write( "treat that potion(S) of the cone as a graph whose shadow D on the xy-plane is the disk of radius 8.  To convert from dS to dA, we need to calculate<BR>\n" );
document.write( "sqrt(1 + (partial derivative z / partial derivative x)^2 + (partial derivative z / partial derivative y)^2) = sqrt(1 + (x^2/(x^2+y^2)) + (y^2/(x^2+y^2))) = sqrt(2)<BR>\n" );
document.write( "therefore the surface area is a double integral<BR>\n" );
document.write( "integrate t from 0 to 2pi integrate r from 0 to 8 sqrt(2)*dr*dt=<BR>\n" );
document.write( "64*pi*sqrt(2)<BR>\n" );
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document.write( "alternately,<BR>\n" );
document.write( "We can unwrap the cone to a sector of a circular disk, with radius 8*sqrt(2) and outer circumference 16*pi (compared to 16*pi*sqrt(2) for the whole circle), so the surface area is pi(8*sqrt(2))^2 / sqrt(2) = 64*pi*sqrt(2)  </SPAN>\n" );
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