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Cone cut by planes parallel to the base: volume ratios by cubes of heights from the vertex

This type of question, asked 6 times: 18 Sep 2025 S1, 18 Sep 2025 S2, 18 Sep 2025 S3, 19 Sep 2025 S1, 19 Sep 2025 S3, 26 Sep 2025 S1. All 6 questions below, oldest first.

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  1. Question 1

    18 Sep 2025 S1

    63
    Creating a frustum, a right circular cone is sliced parallel to its base,. The height of the original cone is 30 cm, and the cut is made 10 cm from the vertex. What is the ratio of the volumes of the frustum to that of the entire cone?

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  2. Question 2

    18 Sep 2025 S2

    63
    A cone is sliced into two sections by a plane that is parallel to its base, with the upper section representing 18\frac{1}{8} of the overall volume. What is the ratio of the heights of the smaller cone to the original cone?

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  3. Question 3

    18 Sep 2025 S3

    63
    A right circular cone is divided into two sections by a plane that is parallel to the base, with the volumes of the frustum and the smaller cone having a ratio of 7:1. Determine the ratio of their heights.

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  4. Question 4

    19 Sep 2025 S1

    63
    A conical bucket is cut at one-third its height from the top. What is the ratio of the volumes of the two parts (top part to bottom part)?

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  5. Question 5

    19 Sep 2025 S3

    62
    A right circular cone having height of 30 cm is cut by two parallel planes at heights 10 cm and 20 cm from the base. What is the ratio of the volumes of the three parts (from top to bottom)?

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  6. Question 6

    26 Sep 2025 S1

    64
    A cone of radius 9 cm and height 12 cm is cut into two parts by a plane parallel to its base, at a height of 4 cm from the vertex. Find the volume of the frustum formed.

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