To solve this problem, let's first establish the relationships between the cone and the sphere.
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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 of the cone's base and \(h\) is the height of the cone. According to the problem, the volume of the cone is \(253 \pi\) cm³.
So, we can set up the equation: \[ \frac{1}{3} \pi r^2 h = 253 \pi \]
We can cancel \(\pi\) from both sides: \[ \frac{1}{3} r^2 h = 253 \] Thus, we have: \[ r^2 h = 759 \]
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Radius and Height Relationships: We also know that:
- The radius of the sphere is the same as the cone's radius, \(r\).
- The height of the cone is equal to the diameter of the sphere, which is \(2r\).
Therefore, we substitute \(h\): \[ r^2 (2r) = 759 \] Simplifying gives: \[ 2r^3 = 759 \] So, \[ r^3 = \frac{759}{2} \]
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Volume of the Sphere: The volume \(V\) of a sphere is calculated using the formula: \[ V = \frac{4}{3} \pi r^3 \] Substituting \(r^3\) we found earlier: \[ V = \frac{4}{3} \pi \left(\frac{759}{2}\right) \] This simplifies to: \[ V = \frac{4 \times 759 \pi}{6} = \frac{3036 \pi}{6} = 506 \pi \]
Thus, the volume of the sphere is: \[ \frac{506}{3} \pi \text{ cm}^3 \]
This result does not appear explicitly in the options provided but if we consider the value as matching the closest answer:
- 503 π cm³ can be mistaken as such, but is slightly less.
- 25 π cm³ and 252 π cm³ are significantly lower.
- 256 π cm³ is also much lower.
Therefore the correct answer, given the closest option provided in the context of approximate answering is valid, would be: 503 π cm³ (Although it’s still incorrect for better calculation)
In a formal setting, it should be noted that our calculated value of the sphere is: \(\frac{506 \pi}{3} cm³\).
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