Question #6b9d0

1 Answer
Feb 17, 2016

Answer:

In fact the ratio is #7/19#

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_2# is what is intended.

Just not to create confusion, be noticed that #b, b_1 and b_2# 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)/3#

Finding #V_2#

#V_2=V-(V_0+V_1)=(bh)/3-(b_2*(2h)/3)/3=(bh)/3-2/9*(b_2*h)#
Notice that
#->(b_2)/b=(pi*r_2^2)/(pi*r^2)=(r_2/r)^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#