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Cone, hemisphere and cylinder on the same base: compare volumes

This type of question, asked 3 times: CDS 1 2021, CDS 2 2024, CDS 2 2025. All 3 questions below, oldest first.

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  1. Question 1

    CDS 1 2021

    61
    A cone and a hemisphere have equal bases and equal volumes. What is the ratio of the height of the cone to the radius of the hemisphere?

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  2. Question 2

    CDS 2 2024

    87
    A cone, a hemisphere and a cylinder stand on equal base of radius rr and have the same height. If the sum of volumes of cone, the hemisphere and the cylinder is equal to volume of a sphere of radius RR, then what is R3r3\frac{R^3}{r^3} equal to ?

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  3. Question 3

    CDS 2 2025

    78
    A right circular cone and a hemisphere have equal base and equal volume. What is the ratio of the height of the cone to the radius of the hemisphere?

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