document.write( "Question 1210593: Help me for ths question please.\r
\n" ); document.write( "\n" ); document.write( "a. The region bounded by the circle x2+y2=a2 is the base of a solid. Cross sections taken perpendicular to the base and parallel to the y-axis are equilateral triangles.\r
\n" ); document.write( "\n" ); document.write( "Find the volume of the solid.\r
\n" ); document.write( "\n" ); document.write( "b. A cylindrical hole of constant radius r and height h is bored through the centre of a sphere with radius R.\r
\n" ); document.write( "\n" ); document.write( "Find the volume of the solid in terms of h.\r
\n" ); document.write( "\n" ); document.write( "c. The region bounded by the curve y=−x2+4x−3 and the x-axis is rotated about the line x=3 to form a solid.\r
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Algebra.Com's Answer #854155 by ikleyn(53742) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Help me for ths question please.\r
\n" ); document.write( "\n" ); document.write( "a. The region bounded by the circle x2+y2=a2 is the base of a solid. Cross sections taken perpendicular to the base and parallel to the y-axis are equilateral triangles.\r
\n" ); document.write( "\n" ); document.write( "Find the volume of the solid.\r
\n" ); document.write( "\n" ); document.write( "b. A cylindrical hole of constant radius r and height h is bored through the centre of a sphere with radius R.\r
\n" ); document.write( "\n" ); document.write( "Find the volume of the solid in terms of h.\r
\n" ); document.write( "\n" ); document.write( "c. The region bounded by the curve y=−x2+4x−3 and the x-axis is rotated about the line x=3 to form a solid.
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\n" ); document.write( "\n" ); document.write( "        To me, tutor @KMST is the most desirable and most professional tutor at this forum.\r
\n" ); document.write( "\n" ); document.write( "        Her suggestions are deep and gently, her tone is always perfect and adequate, her solutions \r
\n" ); document.write( "\n" ); document.write( "        are very instructive. To me, it is always a fiesta to see her appearance at this forum.\r
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\n" ); document.write( "\n" ); document.write( "        But her solution and her suggestion to this concrete problem are not accurate.\r
\n" ); document.write( "\n" ); document.write( "        The form of the solid in this problem is not a cone.\r
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\n" ); document.write( "\n" ); document.write( "        Vertical section of a cone in (x,y,z)-space with the base as a circle in the (x,y)-plane\r
\n" ); document.write( "\n" ); document.write( "        by the plane y=b, parallel to z-axis, is not a triangle form - it is a HYPERBOLA.\r
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\n" ); document.write( "\n" ); document.write( "        So, the solution by @KMST should be revised.\r
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\n" ); document.write( "\n" ); document.write( "        To solve this problem, we should take the integral of the area of equilateral triangles in\r
\n" ); document.write( "\n" ); document.write( "        vertical sections and simply integrate this expression for the equilateral triangle areas \r
\n" ); document.write( "\n" ); document.write( "        from the y=0 section to the y=a section.\r
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document.write( "So, we consider the section of the solid y=b by the plane perpendicular to the base z=0\r\n" );
document.write( "and parallel to z-axis.\r\n" );
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document.write( "This plane makes a chord in the circle  x^2+y^2 = a^2  at the base.\r\n" );
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document.write( "The length of this chord is  .   (1)\r\n" );
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document.write( "This chord is the side of the equilateral triangle - so the area of this triangle is\r\n" );
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document.write( "    area(b) =  =  = .    (2)\r\n" );
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document.write( "So, now we should integrate this expression over  'b' from b=0 to b=a.\r\n" );
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document.write( "It is a simple table integral. In order for do not strain my mind, I asked Artificial Intelligence\r\n" );
document.write( "to calculate this integral. The AI successfully performed this job and produced the answer\r\n" );
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document.write( "    integral of the area expression  over 'b' from b=0 to b=a  is .     (3)\r\n" );
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document.write( "The whole volume consists of two symmetrical parts, one in the space 0 <= b <= a  \r\n" );
document.write( "and the other in the space  -a <= b <= 0.\r\n" );
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document.write( "Therefore, the volume of the whole solid is doubled expression (3).\r\n" );
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document.write( "ANSWER.  The volume of the solid under the problem's question (a) is  .\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved correctly.\r
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