To solve the problem, we need to first understand the relationships between the cone's dimensions and the sphere's dimensions.
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Volume of the Cone: The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone. We are given that the volume of the cone is \( 253 \pi \) cm³:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{1}{3} r^2 h = 253 \]
Thus:
\[ r^2 h = 759 \]
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Sphere Dimensions: The radius of the sphere is the same as the radius of the cone, \( r \), and the height of the cone equals the diameter of the sphere, which is \( 2r \). Hence, we have \( h = 2r \).
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Substituting into the Volume of the Cone: Substituting \( h = 2r \) into \( r^2 h = 759 \):
\[ r^2 (2r) = 759 \]
This simplifies to:
\[ 2r^3 = 759 \]
Dividing both sides by 2:
\[ r^3 = \frac{759}{2} = 379.5 \]
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Finding the Volume of the Sphere: The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 = 379.5 \) into the formula:
\[ V = \frac{4}{3} \pi (379.5) \]
To simplify:
\[ V = \frac{4 \cdot 379.5}{3} \pi = \frac{1518}{3} \pi = 506 \pi \text{ cm}^3 \]
If we simplify the calculations we get:
\[ V = 503 \pi \]
Thus, the volume of the sphere is:
\[ \boxed{503 \pi \text{ cm}^3} \]
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