To find the volume of the sphere, we start by using the volume formula for the cone and the relationship between the cone's dimensions and the sphere's dimensions.
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. According to the problem, the volume of the cone is \( 253 \pi , \text{cm}^3 \). Thus, we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{1}{3} r^2 h = 253 \]
Now, we are also told that the height \( h \) of the cone is equal to the diameter of the sphere. The diameter of the sphere is given by \( d = 2r \), so we can express \( h \) as:
\[ h = 2r \]
Substituting \( h = 2r \) into the cone volume equation:
\[ \frac{1}{3} r^2 (2r) = 253 \]
This simplifies to:
\[ \frac{2}{3} r^3 = 253 \]
Multiplying both sides by \( \frac{3}{2} \):
\[ r^3 = 253 \times \frac{3}{2} = \frac{759}{2} \]
Next, we need to find the volume of the sphere, which is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Now we substitute \( r^3 \) from above into the volume formula of the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{759}{2}\right) \]
This simplifies to:
\[ V_{\text{sphere}} = \frac{4 \cdot 759 \pi}{3 \cdot 2} = \frac{3036 \pi}{6} = \frac{506 \pi}{1} = 506 \pi \]
After calculating the numerical value from \( r^3 \), let’s go back and focus on getting the final volume using:
\[ r^3 = \frac{759}{2}, \] \(\Rightarrow\) using \( \pi \),
Then using:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{759}{2} \right) \]
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1 answer