document.write( "Question 1181731: A sphere is inscribed in a right circular cone of altitude h and radius of base r. Write a formula in terms of r and h for the volume of the sphere. \n" ); document.write( "
Algebra.Com's Answer #850003 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "A sphere is inscribed in a right circular cone of altitude h and radius of base r.
\n" ); document.write( "Write a formula in terms of r and h for the volume of the sphere.
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\n" ); document.write( "\n" ); document.write( "        After attentive reading the solution by @CPhill, I see the errors in it,\r
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\n" ); document.write( "\n" ); document.write( "        So,  I copy the solution by @CPhill and make my editing right there.\r
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document.write( "Here's how to derive the formula for the volume of the inscribed sphere in terms of *r* and *h*:\r\n" );
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document.write( "**1. Diagram and Key Relationships:**\r\n" );
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document.write( "Draw a cross-section of the cone and sphere. You'll see a circle (representing the sphere) inscribed in a triangle (representing the cone).\r\n" );
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document.write( "*   Let *R* be the radius of the inscribed sphere.\r\n" );
document.write( "*   The radius of the cone's base is *r*.\r\n" );
document.write( "*   The height (altitude) of the cone is *h*.\r\n" );
document.write( "*   The slant height of the cone (the hypotenuse of the triangle) is *s* = sqrt(r² + h²).\r\n" );
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document.write( "**2. Similar Triangles:**\r\n" );
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document.write( "There are two similar right triangles in the cross-section:\r\n" );
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document.write( "*   The large triangle representing the cone, with sides *r*, *h*, and *s*.\r\n" );
document.write( "*   A smaller triangle formed by the radius of the sphere (*R*), the difference between the cone's height and the sphere's radius (*h - R*), and a portion of the slant height.\r\n" );
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document.write( "The ratio of corresponding sides in similar triangles is equal:\r\n" );
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document.write( "R / r = (h - R) / s\r\n" );
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document.write( "**3. Solve for R:**\r\n" );
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document.write( "R / r = (h - R) / sqrt(r² + h²)\r\n" );
document.write( "R * sqrt(r² + h²) = r(h - R)\r\n" );
document.write( "R * sqrt(r² + h²) = rh - rR\r\n" );
document.write( "R * sqrt(r² + h²) + rR = rh\r\n" );
document.write( "R(sqrt(r² + h²) + r) = rh\r\n" );
document.write( "R = rh / (sqrt(r² + h²) + r)\r\n" );
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document.write( "**4. Rationalize the Denominator (Optional but often preferred):**\r\n" );
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document.write( "Multiply the numerator and denominator by the conjugate of the denominator:\r\n" );
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document.write( "R = rh(sqrt(r² + h²) - r) / ((sqrt(r² + h²) + r)(sqrt(r² + h²) - r))\r\n" );
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document.write( "R = rh(sqrt(r² + h²) - r) / (r² + h² - r²)\r\n" );
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document.write( "R = rh(sqrt(r² + h²) - r) / h     <<<---=== \"highlight%28highlight%28error%29%29\": should be  R = rh(sqrt(r² + h²) - r) / h²\r\n" );
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document.write( "R = r(sqrt(r² + h²) - r)          <<<---=== as a consequence, should be R = r(sqrt(r² + h²) - r)/h\r\n" );
document.write( "                                            otherwise, the 'R's dimension is m^2, instead of 'm'.\r\n" );
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document.write( "**5. Volume of the Sphere:**\r\n" );
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document.write( "The volume *V* of a sphere is given by:\r\n" );
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document.write( "V = (4/3)πR³\r\n" );
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document.write( "Substitute the expression for *R* we derived:\r\n" );
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document.write( "V = (4/3)π[r(sqrt(r² + h²) - r)/h]³     <<<---===  associated correction here\r\n" );
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document.write( "Therefore, the volume of the inscribed sphere in terms of *r* and *h* is:\r\n" );
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document.write( "V = (4/3)πr³(sqrt((r² + h²) - r)/h)³    <<<---===  associated correction here\r\n" );
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\n" ); document.write( "\n" ); document.write( "Now everything is correct.\r
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\n" ); document.write( "\n" ); document.write( "In previous version by @CPhill, even the dimensions of the expressions were wrong.\r
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\n" ); document.write( "\n" ); document.write( "                Regarding the post by @CPhill . . . \r
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\n" ); document.write( "\n" ); document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r
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\n" ); document.write( "\n" ); document.write( "The artificial intelligence is like a baby now. It is in the experimental stage
\n" ); document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r
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\n" ); document.write( "\n" ); document.write( "                It has no feeling of shame - it is shameless.\r
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\n" ); document.write( "\n" ); document.write( "This time, again,  it made an error.\r
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\n" ); document.write( "\n" ); document.write( "Although the @CPhill' solution are copy-paste  Google  AI solutions,  there is one essential difference.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  Google  AI  makes a note at the end of its solutions that  Google  AI  is experimental
\n" ); document.write( "and can make errors/mistakes.\r
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\n" ); document.write( "\n" ); document.write( "All @CPhill' solutions are copy-paste of  Google  AI  solutions, with one difference:
\n" ); document.write( "@PChill never makes this notice and never says that his solutions are copy-paste that of Google.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  @CPhill embarrasses to tell the truth.\r
\n" ); document.write( "\n" ); document.write( "But I am not embarrassing to tell the truth,  as it is my duty at this forum.\r
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\n" ); document.write( "\n" ); document.write( "And the last my comment.\r
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\n" ); document.write( "\n" ); document.write( "When you obtain such posts from @CPhill,  remember,  that  NOBODY  is responsible for their correctness,
\n" ); document.write( "until the specialists and experts will check and confirm their correctness.\r
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\n" ); document.write( "\n" ); document.write( "Without it,  their reliability is  ZERO and their creadability is  ZERO,  too.\r
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\n" ); document.write( "\n" ); document.write( "Dear @CPhill, I will include such comment to every your post, where I will find an error,\r
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\n" ); document.write( "\n" ); document.write( "UNTIL you will place a correct acknowledgment in every your post.\r
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