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 ?

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, = s²·h

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

∴ V₁ = The area not taken up by the square 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, = π·r²·h

The volume of the 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

=

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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