The volume of a cone is 25 / 3 in cm3 what is the volume of a sphere if it's radius is the same as the cones and the height of the cone is equal to the spirit diameter
To find the volume of the sphere with the same radius as the cone, we first need to determine the radius of the cone from the given volume.
The formula for the volume of a cone is:
\[ V_{cone} = \frac{1}{3}πr^2h \]
where \( V_{cone} \) is the volume of the cone, \( r \) is the radius, and \( h \) is the height. We're given that the volume \( V_{cone} \) is \( \frac{25}{3} \) cubic inches. Let's rearrange the formula and solve for the radius \( r \) in terms of the height \( h \):
\[ \frac{25}{3} = \frac{1}{3}πr^2h \]
\[ 25 = πr^2h \]
Given that the height of the cone is equal to the diameter of the sphere, we have \( h = 2r \), since the diameter is twice the radius. Let's substitute \( h \) with \( 2r \) in the equation:
\[ 25 = πr^2(2r) \]
\[ 25 = 2πr^3 \]
Now, solve for \( r \):
\[ r^3 = \frac{25}{2π} \]
\[ r = \sqrt[3]{\frac{25}{2π}} \]
Now that we have \( r \), we can use it to find the volume of the sphere using the formula:
\[ V_{sphere} = \frac{4}{3}πr^3 \]
Let's calculate \( V_{sphere} \) using the value for \( r \) that we have just found:
\[ V_{sphere} = \frac{4}{3}π\left(\sqrt[3]{\frac{25}{2π}}\right)^3 \]
\[ V_{sphere} = \frac{4}{3}π\left(\frac{25}{2π}\right) \]
\[ V_{sphere} = \frac{4}{3} \times \frac{25}{2} \]
\[ V_{sphere} = \frac{100}{6} \]
\[ V_{sphere} = \frac{50}{3} \text{ cm}^3 \]
So, the volume of the sphere is \( \frac{50}{3} \) cubic inches.