The volume of a sphere is 72 m^3. 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 the cone, we first need to determine the radius of the sphere using 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 \( 72 , \text{m}^3 \): \[ \frac{4}{3} \pi r^3 = 72 \] To solve for \( r^3 \), we first multiply both sides by \( \frac{3}{4} \): \[ \pi r^3 = 72 \cdot \frac{3}{4} = 54 \] Now, we divide by \( \pi \): \[ r^3 = \frac{54}{\pi} \]

To find the radius \( r \): \[ r = \sqrt[3]{\frac{54}{\pi}} \]

Next, we need to determine the height of the cone. The height of the cone is equal to the diameter of the sphere, which is given by: \[ \text{Diameter} = 2r \]

The formula for the volume \( V \) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \): \[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]

Now we can use the value of \( r^3 \) we found earlier: \[ V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right) = \frac{2 \times 54}{3} = \frac{108}{3} = 36 \]

Thus, the volume of the cone is: \[ \boxed{36 , \text{m}^3} \]