To solve the problem, we need to first determine the radius and height of the cone based on the given volume, then we will use that information to find the volume of the sphere.
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The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
We know that the volume of the cone is \( 253 \pi , \text{cm}^3 \). Therefore, we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{1}{3} r^2 h = 253 \]
Multiplying both sides by 3:
\[ r^2 h = 759 \]
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From the problem, we are told that the height of the cone \( h \) is equal to the diameter of the sphere. Since the radius of the sphere \( r \) is the same as the radius of the cone, we have:
\[ h = 2r \]
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Now substituting \( h = 2r \) into the equation \( r^2 h = 759 \):
\[ r^2 (2r) = 759 \]
Simplifying this:
\[ 2r^3 = 759 \]
Dividing both sides by 2:
\[ r^3 = \frac{759}{2} = 379.5 \]
Taking the cube root to find \( r \):
\[ r = \sqrt[3]{379.5} \]
Now we calculate \( r \):
\[ r \approx 7.32 , \text{cm} \quad (\text{using a calculator}) \]
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The sphere's diameter is \( 2r \):
\[ h = 2r \approx 2 \times 7.32 \approx 14.64 , \text{cm} \]
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The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 \approx 379.5 \):
\[ V \approx \frac{4}{3} \pi (379.5) \]
Now calculating \( V \):
\[ V \approx \frac{4}{3} \times 3.14159 \times 379.5 \approx 1595.69 \pi , \text{cm}^3 \]
Thus, the volume of the sphere is approximately \( 1595.69 \pi , \text{cm}^3 \).