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The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the volume of the cone is 253π cm^3, and the height of the cone is equal to the sphere's diameter, we can express the volume of the cone in terms of the sphere's radius (which is also the cone's radius) and height:
253π = (1/3)π(r^2)(2r)
Simplifying, we get:
253 = (2/3)r^3
To find the volume of the sphere with the same radius as the cone, we use the formula V = (4/3)πr^3.
V = (4/3)π(r^3) = (4/3)π (253) ≈ 337π cm^3
Therefore, the volume of the sphere with the same radius as the cone and the height equal to the sphere's diameter is approximately 337π cm^3.