The volume of a cone is 253 π cm3. What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses
25 π cm3
25 pi cm cubed
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
503 π cm3
The volume of a cone is given by the formula:
\[ V_{cone} = \frac{1}{3}\pi r^2 h \]
Where \( V_{cone} \) is the volume of the cone, \( r \) is the radius, and \( h \) is the height. We know the volume of the cone to be \( 253\pi \) cm³.
The radius of the cone is the same as the radius of the sphere, and the height of the cone is equal to the diameter of the sphere. Therefore, the height of the cone is \( 2r \), where \( r \) is the radius of the sphere (and the cone).
To find the radius of the cone (and thus the sphere), we can rearrange the cone volume formula:
\[ 253\pi = \frac{1}{3}\pi r^2 (2r) \]
\[ 253\pi = \frac{2}{3}\pi r^3 \]
\[ 253 = \frac{2}{3}r^3 \]
\[ r^3 = \frac{3 \times 253}{2} \]
\[ r^3 = \frac{759}{2} \]
\[ r^3 = 379.5 \]
The volume of a sphere is given by the formula:
\[ V_{sphere} = \frac{4}{3}\pi r^3 \]
Given that \( r^3 \) is 379.5, the volume of the sphere is:
\[ V_{sphere} = \frac{4}{3}\pi (379.5) \]
\[ V_{sphere} = \pi (4 \times 379.5 / 3) \]
\[ V_{sphere} = \pi (506) \]
\[ V_{sphere} = 506\pi \]
However, none of the provided options match this result. There seems to be an error either in the calculation or the provided options. If we assume that the given volume of the cone is correct, the volume of the sphere using the same radius as calculated should indeed be \( 506\pi \) cm³.