What Fraction of Water Overflows?
Date: 02/04/2002 at 03:54:09 From: Asim Shaukat Subject: Mensuration A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into it and its size is such that when it touches the sides of the conical vessel it is JUST immersed. What fraction of water overflows? I got the correct answer (which is 37.5%) by assuming the radius of the sphere to be half that of the conical vessel. However, I am not able to DERIVE the solution by applying any theories.
Date: 02/04/2002 at 16:39:37 From: Doctor Peterson Subject: Re: Mensuration Hi, Asim. A cross section will show a circle of radius r inscribed in a triangle, with R=6 and H=8: R +----------------oooo+oooo----------------+ \ ooo | ooo / \ oo | oo / \ o | o / \ o r| o / \ o | o / \ o | o / \o + o/ \ | \ r / \ | \ / \ | + \ | / \o H-r| o/ \ooo | ooo/ \ oooo|oooo / \ | / \ | / \ | / \ | / \ | / \|/ + Look for a pair of similar triangles, and use them to write an equation that you can solve for r, which for the particular numbers you have will indeed be R/2, though that would not be true in general. Then find the ratio of the volume of the sphere to the volume of the cone, as you evidently did already. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 02/04/2002 at 22:39:11 From: Asim Shaukat Subject: Mensuration Dr. Peterson, I appreciate the help very much. Thanks and regards, Asim
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