In the figure above, the values in red have been added to the original drawing.
The question asks for the ratio of the left half of the cone to the right half of the cone.
(Please note that, in this context,
half does NOT mean 1/2 the volume.)
The two cones are similar.
It is given that AB=BC.
Thus, AC = 2AB.
Since the height of the larger cone is twice the height of the smaller cone, every dimension of the larger cone -- including the radius -- is twice the corresponding dimension of the smaller cone.
We need to determine the ratio of the volumes.
V = (1/3)�r²h.
Since each volume includes a factor of 1/3 and a factor of �, the ratio of the volumes is not affected by these factors
Thus, we can determine the ratio of the volumes by considering only the two other factors in each volume: r²h.
We can plug in values for the two radii and the two heights.
Smaller cone (the right half):
Let r=1 and h=1.
r²h = (1²)(1)= 1.
Larger cone:
Since each dimension is twice as long, r=2 and h=2.
r²h = (2²)(2)= 8.
Left half of the cone:
Larger cone - smaller cone = 8-1 = 7.
(right half) : (left half) = 1:7.
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