To solve the problem, we need to establish the relationship between the cone and the sphere based on the given information.
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Volume of the cone: The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] where \(r\) is the radius of the cone's base and \(h\) is the height of the cone.We are given that the volume of the cone is \(25 , \text{cm}^3\): \[ \frac{1}{3} \pi r^2 h = 25 \]
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Relationship of the sphere: The radius of the sphere is the same as the radius of the cone, which we denote as \(r\). The height of the cone is equal to the diameter of the sphere, which is \(2r\).
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Substituting \(h = 2r\) into the cone's volume formula:
Replacing \(h\) in the cone's volume formula gives: \[ \frac{1}{3} \pi r^2 (2r) = 25 \] \[ \frac{2}{3} \pi r^3 = 25 \] -
Solve for \(r^3\): Multiplying both sides by \(3/2\): \[ \pi r^3 = 25 \cdot \frac{3}{2} = 37.5 \] \[ r^3 = \frac{37.5}{\pi} \]
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Volume of the sphere: The formula for the volume of a sphere is: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] Now we substitute \(r^3\): \[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{37.5}{\pi}\right) \] Simplifying this: \[ V_{\text{sphere}} = \frac{4}{3} \cdot 37.5 = 50 , \text{cm}^3 \]
Therefore, the volume of the sphere is \( \boxed{50 , \text{cm}^3} \).