document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205289>Question 1205289</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cone of height 9cm has a volume of n cm³ and a curved surface area of n cm², find the vertical angle </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841995 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=841995\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=math_tutor2020&illustration_name=UploadedScreenshot_2.png\"><BR>\n" );
document.write( "V = volume of the cone<BR>\n" );
document.write( "SA = surface area<BR>\n" );
document.write( "LSA = lateral surface area = area of just the curved wall or ceiling<BR>\n" );
document.write( "The LSA ignores the area of the flat base.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "r = radius = unknown<BR>\n" );
document.write( "h = height = 9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Formulas to memorize or have on a reference sheet<BR>\n" );
document.write( "V = (1/3)*pi*r^2*h<BR>\n" );
document.write( "LSA = pi*r*sqrt(r^2+h^2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrt(r^2+h^2) is the slant height due to the Pythagorean theorem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plug h = 9 into each to get<BR>\n" );
document.write( "V = 3pi*r^2<BR>\n" );
document.write( "LSA = pi*r*sqrt(r^2+81)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Those items are both equal to a variable n (just with different units of course). <BR>\n" );
document.write( "That allows us to set them equal to one another to solve for variable r.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "V = LSA<BR>\n" );
document.write( "3pi*r^2 = pi*r*sqrt(r^2+81)<BR>\n" );
document.write( "3pi*r^2 - pi*r*sqrt(r^2+81) = 0<BR>\n" );
document.write( "pi*r*(3r - sqrt(r^2+81)) = 0<BR>\n" );
document.write( "pi*r = 0 or 3r - sqrt(r^2+81) = 0<BR>\n" );
document.write( "r = 0 or 3r - sqrt(r^2+81) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Ignore r = 0 since it's a trivial solution. <BR>\n" );
document.write( "We only consider r > 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3r - sqrt(r^2+81) = 0<BR>\n" );
document.write( "3r = sqrt(r^2+81)<BR>\n" );
document.write( "(3r)^2 = (sqrt(r^2+81))^2<BR>\n" );
document.write( "9r^2 = r^2+81<BR>\n" );
document.write( "9r^2-r^2 = 81<BR>\n" );
document.write( "8r^2 = 81<BR>\n" );
document.write( "r^2 = 81/8<BR>\n" );
document.write( "r = sqrt(81/8)<BR>\n" );
document.write( "r = 3.1819805 approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The apex angle, or vertex angle, of the cone is the angle at the very top. <BR>\n" );
document.write( "Half of this angle is denoted as theta in the diagram.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "theta = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > = greek letter often used for angles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on one of the right triangles.<BR>\n" );
document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(theta) = r/h<BR>\n" );
document.write( "tan(theta) = 3.1819805/9<BR>\n" );
document.write( "tan(theta) = 0.3535534<BR>\n" );
document.write( "theta = arctan(0.3535534)<BR>\n" );
document.write( "theta = 19.4712211 degrees approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This doubles to 2*19.4712211 = <font color=red>38.9424422</font> which is also approximate.<BR>\n" );
document.write( "Round that value however needed.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );