Question 1078514: A cone is hafl filled with water . How deep is d water if d diameter is 12cm and the height is 16cm.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! The entire volume of the cone is V = pi/3*R^2*H where R is the radius and H is the height.
Since the cone is half full, the volume of water in the cone is V/2 = pi/6*R^2*H
We need to find the height of water of the half-full cone.
The volume of water is Vh = pi/3*r^2*h, where r = the radius of the occupied volume and h = the height
We have two unknowns, but from similar triangles, we can write r = (R/H)h
Equating the expressions for the volumes we have
pi/6*R^2*H = pi/3*r^2*h
pi/6*R^2*H = pi/3*(R/H)^2*h^3
H^3 = 2*h^3 -> h = (1/2^(1/3))*H
h = 0.7937*H = 12.6992 ~12.7 cm
Note that our result shows that the height of the half-full cone is always the same fraction of the original height, independent of radius.
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