To find the volume of the sphere based on the information provided about the cone, we first need to extract the dimensions of the cone from its volume.
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.
Given the volume of the cone: \[ V = \frac{25}{3} \pi \text{ cm}^3 \] we can set up the equation: \[ \frac{1}{3} \pi r^2 h = \frac{25}{3} \pi \] Dividing both sides by \(\pi\) and multiplying by 3, we get: \[ r^2 h = 25 \]
Next, we are informed that the height of the cone \( h \) is equal to the diameter of the sphere, and the radius of the sphere is equal to the radius of the cone \( r \). Since the diameter of the sphere is \( 2r \), we have: \[ h = 2r \]
Now substituting this relationship \( h = 2r \) into the equation \( r^2 h = 25 \): \[ r^2 (2r) = 25 \] which simplifies to: \[ 2r^3 = 25 \] From this, we can solve for \( r^3 \): \[ r^3 = \frac{25}{2} \]
Now, we can find the volume of the sphere, which is given by the formula: \[ V_s = \frac{4}{3} \pi r^3 \] Substituting \( r^3 \): \[ V_s = \frac{4}{3} \pi \left(\frac{25}{2}\right) \] This simplifies to: \[ V_s = \frac{4 \cdot 25}{3 \cdot 2} \pi = \frac{100}{6} \pi = \frac{50}{3} \pi \]
Thus, the volume of the sphere is: \[ \frac{50}{3} \pi \text{ cm}^3 \]
The answer is: \[ \boxed{\frac{50}{3} \pi \text{ cm}^3} \]
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