The volume of a cone is S × H/3, where S is the area of the base and H

Question

https://gmatclub.com/forum/download/file.php?id=27418 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 . cone-1.gif

VenoMfTw · Accepted Answer

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 cone-1.gif 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

GMATinsight · Answer

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

colorbrandon · Answer

Here's my solution

Answer, D

Bunuel · Answer

800score Official Solution:

1) We can create two similar triangles using AC and AB as the heights, R and x as the bases, and the edge of the cone as the hypotenuse. These triangles are similar because the angles for both are the same. The sides of similar triangles are proportional to each other.

2) Because AB = BC, and AC = AB + BC, then 2(AB) = AC.

3) Since the triangles are similar, and because 2(AB) = AC, 2x = R.

4) The volume of the left half of the cone equals the volume of the entire cone minus the volume of the right half of the cone, so the ratio we are looking for is:
(volume of right half) : (volume of cone – volume of right half).

5) Volume of the entire cone = S × H/3
= (πR²) × (AC)/3
= (π(2x)²) × (2AB)/3
= 8πx²AB/3

6) Volume of the right half of the cone = S × H/3
= πx²AB/3

7) Plug these values into the ratio equation above and solve:
πx²AB/3 : (8πx²AB/3 – πx²AB/3)
πx²AB/3 : 7πx²AB/3 
1 : 7.

The correct answer is D.

mvictor · Answer

if AB is equal to 1/2H then x must be R/2
let's find the volume of the small cone with the base on the B
we know that radius of the small circle is R/2 of the big, and height is 1/2 of the H.
volume must be (R/2)^2 * pi * H/3 * 1/2 = R^2/4 * pi * H * 1/6 or (R^2 * pi * H) / 24
this is the volume of the small cone.

the volume of the cone from B to C is volume of the big A-C cone minus the small cone A->B.
volume of the big cone is R^2*pi * H/3 or (8 * R^2 * pi * H) / 24
subtract from the whole, the small one
(8 * R^2 * pi * H) / 24 minus
(R^2 * pi * H) / 24

we get (7 * R^2 * pi * H) / 24 - volume of the cone B-C

volume of the A->B to the B-C is then:
(R^2 * pi * H) / 24
divided everything by 
(7 * R^2 * pi * H) / 24

or (R^2 * pi * H) * 24 / 24 * (7 * R^2 * pi * H)
simplify by 24, R^2, pi, and H, and remain with 1/7.

tough and tricky one!!!

hellosanthosh2k2 · Answer

One General idea i have,
whenever, we have two similiar triangles with sides in ratio 1:x , 
then ratio of the areas of the triangles will be of ratio 1 : x^2

now extending the concept to 3-D dimension, if there are two similiar cones of side ratio 1:x
then ratio of the volumes of cones will be of ratio 1:x^3

now we have small right cone and parent cone, small cone height/parent cone height = 1:2
so volume of small right cone/volume of parent cone = 1:8

so volume of remaining part of cone/volume of parent cone = 7:8
so volume of small right cone/volume of remaining part of cone = 1:7

Fdambro294 · Answer

Once you understand that the concept regarding Ratio of Lengths and Similar 2-D shapes (such as triangles and squares) also applies to the Volumes of 3-D Figures (albeit a little differently), the question can be answered rather quickly.

Furthermore, the Formula for the Volume of a Cone is:

Volume of Cone = (1/3) * (Volume of corresponding Cylinder)

The given information about the Volume of the Cone is just giving restating this formula, where S = area of circular base and H = height.

Key concept: Given two Similar 3-D figures with corresponding length measures in the Ratio of:

A : B

Then the Ratio of the VOLUMES of these 2 similar figures is:

(A)^3 : (B)^3

Based on the figure drawn, the Smaller/Right cone is Similar to the Entire Cone.

Since we are given that the Top Cone’s Height is equal to (1/2) of the Height of the Entire Cone, the Constant Ratio of corresponding lengths between the 2 Cones is:

(Smaller/Right Cone Height) : (Entire Cone Height) = (1) : (2)

This means the Ratio of Volumes is the following:

(Smaller Cone Volume) : (Entire Cone volume) = (1)^3 : (2)^3 = 1 : 8

The Left portion of the figure is the Difference between the (Volume of Entire Cone) - (Volume of Top, smaller cone) = 8 - 1 = 7

Therefore, the Ratio of the Volumes of the 2 parts of the figure is:

1 : 7

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The volume of a cone is S × H/3, where S is the area of the base and H

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The volume of a cone is S × H/3, where S is the area of the base and H

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