document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181727>Question 1181727</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cone is inscribed in a sphere. It has for its base a great circle of the sphere and for its vertex a pole of that circle. Find the ratio (a) of the total area of the cone to the area of the sphere, (b) of the volume of the cone to the volume of the sphere. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849975 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849975\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the ratios:\r<BR>\n" );
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document.write( "**1. Visualize the Setup:** Imagine a sphere.  The cone's base is a great circle (a circle with the sphere's radius), and the cone's tip (vertex) is at one of the poles (farthest points from the great circle).\r<BR>\n" );
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document.write( "**2. Key Dimensions:**\r<BR>\n" );
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document.write( "*   Let *R* be the radius of the sphere.<BR>\n" );
document.write( "*   The radius of the cone's base (r) is equal to the sphere's radius: r = R.<BR>\n" );
document.write( "*   The height (h) of the cone is also equal to the sphere's radius: h = R.<BR>\n" );
document.write( "*   The slant height (s) of the cone can be found using the Pythagorean theorem: s = sqrt(r² + h²) = sqrt(R² + R²) = R√2\r<BR>\n" );
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document.write( "**3. (a) Ratio of Total Area of Cone to Area of Sphere:**\r<BR>\n" );
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document.write( "*   Area of the cone's base = πr² = πR²<BR>\n" );
document.write( "*   Lateral surface area of the cone = πrs = πR(R√2) = πR²√2<BR>\n" );
document.write( "*   Total surface area of the cone = Area of base + Lateral surface area = πR² + πR²√2 = πR²(1 + √2)<BR>\n" );
document.write( "*   Surface area of the sphere = 4πR²\r<BR>\n" );
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document.write( "Ratio of areas = [πR²(1 + √2)] / (4πR²) = (1 + √2) / 4\r<BR>\n" );
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document.write( "**4. (b) Ratio of Volume of Cone to Volume of Sphere:**\r<BR>\n" );
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document.write( "*   Volume of the cone = (1/3)πr²h = (1/3)πR²R = (1/3)πR³<BR>\n" );
document.write( "*   Volume of the sphere = (4/3)πR³\r<BR>\n" );
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document.write( "Ratio of volumes = [(1/3)πR³] / [(4/3)πR³] = 1/4\r<BR>\n" );
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document.write( "**Summary of Ratios:**\r<BR>\n" );
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document.write( "*   (a) Ratio of total area of cone to area of sphere: (1 + √2) / 4<BR>\n" );
document.write( "*   (b) Ratio of volume of cone to volume of sphere: 1/4<BR>\n" );
document.write( " </SPAN>\n" );
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