To solve the problem, we need to find the radius of the cone using its volume and then use that radius to find the volume of the sphere.
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Volume of the cone: The formula for the volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.
We know from the problem that the volume of the cone is \( 253 \pi \) cm³, so we set up the equation:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
By dividing both sides by \( \pi \), we get:
\[ \frac{1}{3} r^2 h = 253 \]
Reorganizing gives us:
\[ r^2 h = 759 \quad (1) \]
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Relationship between height and radius: According to the problem, the height of the cone is equal to the diameter of the sphere. Since the diameter of the sphere is \( 2r \), we can state:
\[ h = 2r \quad (2) \]
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Substituting (2) into (1): We can now substitute \( h = 2r \) into our expression from step (1):
\[ r^2 (2r) = 759 \]
This simplifies to:
\[ 2r^3 = 759 \]
Dividing both sides by 2, we find:
\[ r^3 = 379.5 \]
Now we will calculate \( r \):
\[ r = (379.5)^{1/3} \approx 7.25 \text{ cm} \]
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Finding the volume of the sphere: The volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Using \( r^3 = 379.5 \):
\[ V = \frac{4}{3} \pi (379.5) \]
Calculating this, we get:
\[ V = \frac{4 \times 379.5}{3} \pi = \frac{1518}{3} \pi = 506 \pi , \text{cm}^3 \]
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Conclusion: The volume of the sphere is \( 506 \pi , \text{cm}^3 \). The closest option listed is:
None of the options in the provided list match. Please check the answer options again or confirm the conditions of the problem.
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