Question

To solve the problem, we need to use the formulas for the volumes of the cylinder, sphere, and relate their dimensions as given.

1. Volume of a cylinder (V_cylinder) is given by: \[ V_{cylinder} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

2. Volume of a sphere (V_sphere) is given by: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]

Given:

• Volume of the cylinder, \( V_{cylinder} = 36 , \text{cm}^3 \)
• The height of the cylinder is equal to the diameter of the sphere. Thus, \( h = 2r \).

Setup:

We can substitute \( h \) into the volume of the cylinder: \[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \] Setting this equal to the given volume of the cylinder: \[ 2\pi r^3 = 36 \]

Solve for \( r^3 \):

\[ \pi r^3 = 18 \quad \Rightarrow \quad r^3 = \frac{18}{\pi} \]

Now, let's find the volume of the sphere:

Substituting \( r^3 \) into the formula for the volume of the sphere: \[ V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) = \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 , \text{cm}^3 \]

Conclusion:

The volume of the sphere is \( 24 , \text{cm}^3 \).

Responses:

24 cm³ 24 cm cubed