To solve this problem, we can start with the formula for the volume of a cone and the volume of a sphere.
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 \( \frac{25}{3} \pi \) cm³. Therefore, we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = \frac{25}{3} \pi \]
Cancelling \( \pi \) from both sides, we have:
\[ \frac{1}{3} r^2 h = \frac{25}{3} \]
Multiplying both sides by 3 gives us:
\[ r^2 h = 25 \]
The problem states that the height of the cone \( h \) is equal to the diameter of the sphere. We know that the diameter \( d \) of the sphere is twice the radius \( r \):
\[ d = 2r \]
So, we can substitute \( h = 2r \) into the equation \( r^2 h = 25 \):
\[ r^2 (2r) = 25 \]
This simplifies to:
\[ 2r^3 = 25 \]
Dividing both sides by 2, we find:
\[ r^3 = \frac{25}{2} \]
Now we need to find the volume of the sphere, which is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 \) from above into this formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{25}{2} \right) \]
Calculating this gives:
\[ V_{\text{sphere}} = \frac{4 \cdot 25}{3 \cdot 2} \pi = \frac{100}{6} \pi = \frac{50}{3} \pi \]
Thus, the volume of the sphere is:
\[ \frac{50}{3} \pi \text{ cm}^3 \]
The correct answer is:
\[ \frac{50}{3} \pi \text{ cm}^3 \]
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