Question

The volume of a sphere is 72 m3
. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?

Answer 1

To find the volume of a cone with the same radius as a sphere and a height equal to the sphere's diameter, 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 the volume of the sphere is \( 72 \, \text{m}^3 \), we can set up the equation:

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

To solve for \( r^3 \), we will multiply both sides by \( \frac{3}{4\pi} \):

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

Now, we need to find \( r \):

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

Next, we calculate the diameter of the sphere, which is:

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

Now that we have established the radius and height of the cone, we can use the formula for the volume \( V \) of a cone:

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

Here, \( h = 2r \). Substituting this into the formula gives:

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

Now, we can substitute \( r^3 \) (which we found earlier):

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

This simplifies to:

\[
V = \frac{2 \cdot 54}{3} = \frac{108}{3} = 36 \, \text{m}^3
\]

Thus, the volume of the cone is:

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