The volume of a sphere is 72 m^3. What is the volume of the cone if its radius is same as the sphere's and its height is equal to the sphere's diameter.

1 answer

To find the volume of the cone, we first need to determine the radius of the sphere from its given volume.

The formula for the volume \( V \) of a sphere is:

\[ V = \frac{4}{3} \pi r^3 \]

Given that \( V = 72 , \text{m}^3 \), we can set up the equation:

\[ \frac{4}{3} \pi r^3 = 72 \]

To solve for \( r^3 \), we'll isolate \( r^3 \):

\[ r^3 = \frac{72 \cdot 3}{4\pi} = \frac{216}{4\pi} = \frac{54}{\pi} \]

Next, we want to find the radius \( r \):

\[ r = \left(\frac{54}{\pi}\right)^{1/3} \]

Now, the diameter \( d \) of the sphere, which is twice the radius, is:

\[ d = 2r = 2\left(\frac{54}{\pi}\right)^{1/3} \]

The formula for the volume \( V \) of a cone is:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius of the cone (which is the same as the sphere's radius) and \( h \) is the height of the cone (which is equal to the diameter of the sphere). Therefore, we substitute \( h = d = 2r \):

\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we substitute \( r^3 \) from our earlier calculations:

\[ V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right) \]

This simplifies as follows:

\[ V = \frac{2}{3} \cdot 54 = \frac{108}{3} = 36 \]

So, the volume of the cone is:

\[ \boxed{36 , \text{m}^3} \]