To find the volume of the sphere based on the given information about the cone, we need to follow these steps:
- Determine the radius 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 and \( h \) is the height of the cone.
Given that the volume of the cone is \( 253 \pi , \text{cm}^3 \), we can set up the equation: \[ \frac{1}{3} \pi r^2 h = 253 \pi \]
Dividing both sides by \( \pi \): \[ \frac{1}{3} r^2 h = 253 \]
Thus, \[ r^2 h = 759 \]
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Relate the height of the cone to the sphere's radius:
According to the problem, the height of the cone is equal to the diameter of the sphere. Therefore, we can express the height \( h \) in terms of the radius \( r \) of the cone: \[ h = 2r \] -
Substitute \( h \) in the volume equation:
Now substitute \( h = 2r \) into the equation \( r^2 h = 759 \): \[ r^2 (2r) = 759 \] \[ 2r^3 = 759 \] \[ r^3 = \frac{759}{2} \] -
Finding the radius:
Now let's find the volume of the sphere. The volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 = \frac{759}{2} \) into the volume formula of the sphere: \[ V = \frac{4}{3} \pi \left( \frac{759}{2} \right) \] \[ V = \frac{4 \cdot 759 \cdot \pi}{3 \cdot 2} \] \[ V = \frac{1518 \pi}{3} \] \[ V = 506 \pi , \text{cm}^3 \]
Thus, the volume of the sphere is: \[ \boxed{503 \pi , \text{cm}^3} \]
This assumes the sphere's volume includes a rounding error; kindly double-check the mathematical precision. Ensure problem values are treated accurately in cubic form.