Question #6b9d0

1 Answer
Feb 17, 2016

In fact the ratio is 7/19719

Explanation:

Refer to the figure below

I created this figure using MS Excel

Since the 2 planes divide the cone in 3 parts we can obtain 3 ratios. But I presume that V_1/V_2V1V2 is what is intended.

Just not to create confusion, be noticed that b, b_1 and b_2b,b1andb2 are the areas (in square units) of the bases of their respective cones.

The volume of a cone is given as
V=(S_(base)*height)/3V=Sbaseheight3

Finding V_2V2

V_2=V-(V_0+V_1)=(bh)/3-(b_2*(2h)/3)/3=(bh)/3-2/9*(b_2*h)V2=V(V0+V1)=bh3b22h33=bh329(b2h)
Notice that
->(b_2)/b=(pi*r_2^2)/(pi*r^2)=(r_2/r)^2b2b=πr22πr2=(r2r)2 and since r_2/r=((2cancel(h))/3)/cancel(h)=2/3
=>(b_2)/b=(2/3)^2 => b_2=4/9*b
So
V_2=(bh)/3-2/9*(b_2*4/9b)=(27bh-8bh)/81 => V_2=(19*bh)/81

Finding V_1

V_1=V-V_0-V_2=(bh)/3-(b_1*h/3)/3-(19*bh)/81
But as we saw above
b_1/b=((cancel(h)/3)/cancel(h))^2=(1/3)^2 => b_1=b/9
So
V_1=(bh)/3-b/9*h/9-(19*bh)/81=(27*bh-bh-19*bh)/81 => V_1=(7*bh)/81

So the asked ratio is

V_1/V_2=((7*cancel(bh))/cancel(81))/((19*cancel(bh))/cancel(81))=7/19