To find the volume of the sphere given the volume of the cone, we need to establish the relationship between the dimensions of the cone and the sphere.
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Volume of the cone formula: The volume \( V \) of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
From the problem, we know: \[ \frac{25}{3} \pi = \frac{1}{3} \pi r^2 h \]
By eliminating \( \frac{1}{3} \pi \) from both sides, we have: \[ 25 = r^2 h \]
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Height of the cone and diameter of the sphere: The problem states that the height of the cone is equal to the diameter of the sphere. The diameter of the sphere \( d \) is related to the radius \( r \) of the sphere by: \[ d = 2r \] Thus, we can set the height of the cone \( h \) equal to the diameter: \[ h = 2r \]
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Substituting for \( h \): We substitute \( h \) into the equation \( 25 = r^2 h \): \[ 25 = r^2 (2r) \] Simplifying gives: \[ 25 = 2r^3 \implies r^3 = \frac{25}{2} \implies r = \left(\frac{25}{2}\right)^{\frac{1}{3}} \]
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Finding the radius: Instead of calculating \( r \) explicitly, we use the relationship further on to find the volume of the sphere.
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Volume of the sphere formula: The volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \]
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Calculating \( r^3 \): We derived that \( r^3 = \frac{25}{2} \).
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Substituting \( r^3 \) into the volume formula: Now substituting \( r^3 \) into the volume formula for the sphere: \[ V = \frac{4}{3} \pi \left(\frac{25}{2}\right) \] Simplifying gives: \[ V = \frac{4 \cdot 25}{3 \cdot 2} \pi = \frac{100}{6} \pi = \frac{50}{3} \pi \]
Therefore, the volume of the sphere is \( \frac{50}{3} \pi \) cm³.
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