document.write( "Question 890514: The question is as under :\r
\n" ); document.write( "\n" ); document.write( "\" The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.\"\r
\n" ); document.write( "\n" ); document.write( "The above question is related to Maxima and Minima chapter of Differential calculus.\r
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\n" ); document.write( "\n" ); document.write( "Khoka123. \n" ); document.write( "
Algebra.Com's Answer #539037 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You are to prove it in general for any given value of the total surface\r\n" );
document.write( "area,  So we just call it a constant A.\r\n" );
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document.write( "Since the problem mentions diameter rather than radius, let's\r\n" );
document.write( "convert the standard formula for the volume of a sphere so that\r\n" );
document.write( "it is in terms of the diameter D instead of the radius, so we\r\n" );
document.write( "substitute \"D%2F2\" for r in the usual formula:\r\n" );
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document.write( "And also for the surface area of a sphere\r\n" );
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document.write( "\"A=4pi%2Ar%5E2=4pi%2A%28D%2F2%29%5E2=4pi%2A%28D%5E2%2F4%29=pi%2AD%5E2\"\r\n" );
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document.write( "\"A-pi%2AD%5E2\"\"%22%22=%22%22\"\"6x%5E2\"\r\n" );
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document.write( "\"%28A-pi%2AD%5E2%29%2F6\"\"%22%22=%22%22\"\"x%5E2\"\r\n" );
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document.write( "\"sqrt%28%28A-pi%2AD%5E2%29%2F6%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"%28%28A-pi%2AD%5E2%29%2F6%29%5E%281%2F2%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"6%5E%28-1%2F2%29%28A-pi%2AD%5E2%29%5E%281%2F2%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"6%5E%28-3%2F2%29%28A-pi%2AD%5E2%29%5E%283%2F2%29%7D%7D%7B%7B%7B%22%22=%22%22\"\"x%5E3\"\r\n" );
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document.write( "Substitute in:\r\n" );
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document.write( "Put that equal to zero:\r\n" );
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document.write( "Multiply both sides by 2 to clear the fraction:\r\n" );
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document.write( "Add the exponents of 6\r\n" );
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document.write( "Divide by \"pi%2AD\"  (we aren't interested in when D=0)\r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "Multiply both sides by 6 to clear the fraction:\r\n" );
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document.write( "\"matrix%281%2C5%2C%0D%0A%0D%0A6D%5E2%2C%22%22%2C+%22%22=%22%22%2C%22%22%2C+A-pi%2AD%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%281%2C5%2C%0D%0A%0D%0A6D%5E2%2Bpi%2AD%5E2%2C%22%22%2C+%22%22=%22%22%2C%22%22%2C+A%29\"\r\n" );
document.write( "\"matrix%281%2C5%2C%0D%0A%0D%0AD%5E2%286%2Bpi%29%2C%22%22%2C+%22%22=%22%22%2C%22%22%2C+A%29\"\r\n" );
document.write( "\"matrix%281%2C5%2C%0D%0A%0D%0AD%2C%22%22%2C+%22%22=%22%22%2C%22%22%2C+A%2F%286%2Bpi%29%29\"\r\n" );
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document.write( "Take positive square roots of both sides:\r\n" );
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document.write( "Now we only need to show that x also equals that.\r\n" );
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document.write( "\"sqrt%28%28A-pi%2AD%5E2%29%2F6%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "We substitute for D:\r\n" );
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document.write( "\"sqrt%28%28A-pi%2A%28sqrt%28A%2F%286%2Bpi%29%29%29%5E2%29%2F6%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"sqrt%28%28A-pi%2A%28A%2F%286%2Bpi%29%29%29%2F6%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"sqrt%28%28A-%28pi%2AA%2F%286%2Bpi%29%29%29%2F6%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "Multiply top and bottom under the radical by \"%286%2Bpi%29\"\r\n" );
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document.write( "\"sqrt%28%28A%286%2Bpi%29-pi%2AA%29%2F%286%286%2Bpi%29%29%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"sqrt%28%286A%2Bpi%2AA-pi%2AA%29%2F%286%286%2Bpi%29%29%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"sqrt%28%286A%29%2F%286%286%2Bpi%29%29%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "\"sqrt%28A%2F%286%2Bpi%29%29\"\"%22%22=%22%22\"\"x\"\r\n" );
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document.write( "So x = r and the proposition is true.\r\n" );
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document.write( "Edwin
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