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A cone has been divided into four slices parallel to the base.

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vaibhav1221
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A cone has been divided into four slices parallel to the base. [#permalink] New post   Updated on: 23 Jan 2022, 04:26
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A cone has been divided into four slices parallel to the base. The heights of the various slices from top to bottom are in the ratio 1 : 2 : 3 : 4. If the volume of the bottommost slice is 784 cm^3, what is difference in volumes of the third slice from the top and the second slice from the top (in cm^3)?

A. 576
B. 196
C. 729
D. 125
E. 163


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Originally posted by vaibhav1221 on 22 Jan 2022, 01:15.
Last edited by Bunuel on 23 Jan 2022, 04:26, edited 3 times in total.
Edited the question.
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Re: A cone has been divided into four slices parallel to the base. [#permalink] New post   12 Feb 2022, 05:49
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There is only one concept we need to know here - the rest if just plain arithmetic/algebra. The ratio of volumes of two similar bodies is proportional to the ratio of the cubes of their corresponding sides.

    Ratio of heights of the four segments = 1:2:3:4
    Ratio of heights of the four cones (from the top) = 1:3:6:10 (Add the heights of the previous segments)
    Then, the corresponding ratios of their volumes will be: 1^3:3^3:6^3:10^3, i.e. 1:27:216:1000

Let's assume the volumes are 1V, 27V, 216V and 1000V. We are given that the fourth slice = 784 cm^3. So, 1000V - 216V = 784. Therefore, we get V = 1 cm^3.

Plugging the values, we get volume of third slice = 216V - 27V = 189 (We need to subtract the volume of the preceding cone to get the volume of a given slice)
volume of second slice = 27V - 1V = 26

So, the difference in volumes = 189 - 26 = E) 163

cc: [b]Bunuel, vaibhav1221
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Re: A cone has been divided into four slices parallel to the base. [#permalink] New post   14 Jan 2024, 12:36
I am confused because the radius would be different in addition to the heights. We did not account for the difference in radius.
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Re: A cone has been divided into four slices parallel to the base. [#permalink] New post   21 Jan 2024, 05:37
anud33p wrote:
There is only one concept we need to know here - the rest if just plain arithmetic/algebra. The ratio of volumes of two similar bodies is proportional to the ratio of the cubes of their corresponding sides.

    Ratio of heights of the four segments = 1:2:3:4
    Ratio of heights of the four cones (from the top) = 1:3:6:10 (Add the heights of the previous segments)
    Then, the corresponding ratios of their volumes will be: 1^3:3^3:6^3:10^3, i.e. 1:27:216:1000

Let's assume the volumes are 1V, 27V, 216V and 1000V. We are given that the fourth slice = 784 cm^3. So, 1000V - 216V = 784. Therefore, we get V = 1 cm^3.

Plugging the values, we get volume of third slice = 216V - 27V = 189 (We need to subtract the volume of the preceding cone to get the volume of a given slice)
volume of second slice = 27V - 1V = 26

So, the difference in volumes = 189 - 26 = E) 163

cc: [b]Bunuel, vaibhav1221


Hi Bunuel, can you please help me understand when we are figuring out the volume of the 4th slice, why are we subtracting only 216V from 1000V, why not 1000V-(1V+27V+216V)=756V? Same for the other slices, why do we only subtract the previous slice from the current one?
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Re: A cone has been divided into four slices parallel to the base. [#permalink]
21 Jan 2024, 05:37
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