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29 Jul 2015, 03:27
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D
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Difficulty:
85%
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Question Stats:
48% (02:42) correct
52%
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The volume of a cone is S × H/3, where S is the area of the base and H is the height. According to the figure above, if AB = BC, what is the ratio of the volume of the right section of the cone (with AB as height) to the volume of the left section of the cone (with BC as height)? (Note: Figure not drawn to scale.)
A. 1 : 3
B. 1 : 4
C. 1 : 6
D. 1 : 7
E. 1 : 8
Kudos for a correct solution.
Attachment:
cone-1.gif [ 7 KiB | Viewed 7347 times ]
29 Jul 2015, 04:51
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Bunuel
Let AB = BC = h. Then AC = 2h
Let the line extended from B meet at D and from C meet at E as shown in fig below
Attachment:
cone-1.gif [ 6.52 KiB | Viewed 6912 times ]
Triangle ABD and Triangle ACE are similar triangles ( ∠BAD = ∠ CAE , ∠B = ∠C =90)
Thus \(\frac{AB}{AC}\) = \(\frac{1}{2}\)
And \(\frac{BD}{CE}\) = \(\frac{1}{2}\) ==> \(\frac{x}{R}\) = \(\frac{1}{2}\)
R = 2x
volume of the right section of the cone (with AB as height) = π\(r^2\) * \(\frac{AB}{3}\)
= π\(x^2\) * \(\frac{h}{3}\)
volume of the left section of the cone (with BC as height) = volume of the left section of the cone (with AC as height) - volume of the right section of the cone (with AB as height)
volume of the left section of the cone (with AC as height) = π\(r^2\) * \(\frac{AC}{3}\)
=π\(R^2\) * \(\frac{2h}{3}\)
=π\(2x^2\) * \(\frac{2h}{3}\)
=π\(x^2\) * \(\frac{8h}{3}\)
Thus volume of the left section of the cone (with BC as height) = π\(x^2\) * \(\frac{8h}{3}\) - π\(x^2\) * \(\frac{h}{3}\)
= π\(x^2\) * \(\frac{7h}{3}\)
Thus Ratio = 1:7
General Discussion
Updated on: 29 Jul 2015, 06:09
Originally posted by GMATinsight on 29 Jul 2015, 04:47.
Last edited by GMATinsight on 29 Jul 2015, 06:09, edited 1 time in total.
Last edited by GMATinsight on 29 Jul 2015, 06:09, edited 1 time in total.
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Bunuel
Ratio of Height of smaller and bigger cone = 1/2
CONCEPT: Ratio of Volume of similar solids = (Ratio of Height)^3
So Ratio of Volume of similar solids = (Ratio of Height)^3 = (1/2)^3 = 1/8
i.e. Volume of Right Section (Smaller Cone) = 1
then, Volume of Bigger Cone (including Smaller Cone) = 8
i.e. Volume of Left Section (Frustum) = Volume of Bigger Cone - Volume of Smaller Cone = 8-1 = 7
i.e. volume of Left Section = 8-1 =7
i.e. Volume of Right Section / Volume of Right Section = 1 / 7
Answer: Option D
