document.write( "Question 1189457: A right circular cone with top width 24cm and altitude 8cm is filled with water. A spherical steel ball with radius 3.0cm is submerged in the cone. Find the volume of water below the sphere. \n" ); document.write( "

Algebra.Com's Answer #820951 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Unlike the other tutor, I will assume the ball is heavy \r\n" );
document.write( "enough to sink all the way down and touch the sides of \r\n" );
document.write( "the cone.  No offense, Greenestamps! lol\r\n" );
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document.write( "Here is the way it looks:\r\n" );
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document.write( "It's easy to see that all 6 right triangles DFC, EFC, OPA, OPB, \r\n" );
document.write( "APC, and BPC are similar. Here's a mid-cross-section \r\n" );
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document.write( "By using the Pythagorean theorem to calculate the hypotenuse CD = 4√(13),\r\n" );
document.write( "we see that this ratio holds for all 6 right triangles \r\n" );
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document.write( "shorter leg : longer leg : hypotenuse = 8 : 12 : \r\n" );
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document.write( "Dividing through by 4\r\n" );
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document.write( "shorter leg : longer leg : hypotenuse = 2 : 3 : \r\n" );
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document.write( "Below we will draw the kite ACBO with all the lengths which\r\n" );
document.write( "we calculate by ratio and proportion of similar triangles\r\n" );
document.write( "(You don't need to calculate them all).\r\n" );
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document.write( "So the radius of the cone at the bottom (that the sphere \"sits in\") is \r\n" );
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document.write( "and the height of that cone is\r\n" );
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document.write( "We compute its volume by the formula:\r\n" );
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document.write( "However, we see that the volume we want is not the entire cone, because the cone\r\n" );
document.write( "is \"dished out\" by the sphere sitting on it.  The \"dished out\" part is a cap of\r\n" );
document.write( "the sphere.  So we must calculate the volume of the cap of the sphere which\r\n" );
document.write( "dishes out the cone, so we can subtract it from the above result.  The formula\r\n" );
document.write( "for the volume of the cap of a sphere is\r\n" );
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document.write( "where r = radius of the sphere = 3, and \r\n" );
document.write( "h = height of the cap.\r\n" );
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document.write( "We calculate the height of the cap, PQ:\r\n" );
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document.write( "PQ = OQ - OP.  OQ is a radius of the sphere, so OQ = 3. So\r\n" );
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document.write( "So the volume of the cap is\r\n" );
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document.write( "That simplifies to \r\n" );
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document.write( "So we subtract that from   and get the\r\n" );
document.write( "volume of water below the sphere as\r\n" );
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document.write( "which simplifies to:\r\n" );
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document.write( " cubic centimeters.\r\n" );
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document.write( "Edwin

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