A square pyramid is inscribed in a rectangular prism. A cone is inscribed in a cylinder. The pyramid and the cone have the same volume. Part

Question

A square pyramid is inscribed in a rectangular prism. A cone is inscribed in a cylinder. The pyramid and the cone have the same volume. Part of the volume of the rectangular prism, 1 V 1 , is not taken up by the square pyramid. Part of the volume of the cylinder, 2 V 2 , is not taken up by the cone. What is the relationship of these two volumes, 1 V 1 and 2 V 2 ?

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Gerda 5 years 2021-07-25T16:37:03+00:00 1 Answers 182 views 1

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  1. Dulcie
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    2021-07-25T16:38:23+00:00
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    Answer:

    V₂ = V₁

    Step-by-step explanation:

    Let the height of the rectangular prism = h

    Let s represent the side length of the base of the square prism, we have;

    The volume of the prism, V_{prism} = s²·h

    The volume of the square pyramid, V_{pyramid} = (1/3)·s²·h

    ∴ V₁ = The area not taken up by the square pyramid = V_{prism} – V_{pyramid}

    ∴ V₁ = s²·h – (1/3)·s²·h = (2/3)·s²·h

    Similarly, for the cylinder, we have;

    Let h represent the height of the cylinder

    Let r represent the radius of the base of the cone, we have;

    Therefore;

    The volume of the cylinder, V_{cylinder} = π·r²·h

    The volume of the cone, V_{cone} = (1/3)·π·r²·h

    ∴ V₂ = π·r²·h – (1/3)·π·r²·h = (2/3)·π·r²·h

    V₂ = (2/3)·π·r²·h

    V_{cone} = V_{pyramid}

    Therefore;

    (1/3)·π·r²·h = (1/3)·s²·h

    ∴ π·r² = s²

    Therefore, V₂ = (2/3)·π·r²·h = V₂ = (2/3)·s²·h = V₁

    V₂ = V₁.

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