Calculate Number of Balls from Solid Sphere

The problem asks for the number of smaller spheres (balls) that can be formed from a larger solid sphere. This can be determined by comparing the volumes of the two spheres.

Sphere Volume Formula

The volume ($V$) of a sphere is calculated using the formula:

$V = \frac{4}{3}\pi r^3$

where $r$ is the radius of the sphere.

Volume Comparison

Let $R$ denote the radius of the large solid sphere and $r$ denote the radius of the smaller balls.

• Given $R = 4$ cm.
• Given $r = 2$ cm.

The volume of the large sphere is:

$V_{large} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (4)^3 \text{ cm}^3$

The volume of one small ball is:

$V_{small} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (2)^3 \text{ cm}^3$

Number of Balls Calculation

The total number of small balls is the ratio of the volume of the large sphere to the volume of a small ball:

$ \text{Number of balls} = \frac{V_{large}}{V_{small}} $

$ \text{Number of balls} = \frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3} $

The $\frac{4}{3}\pi$ terms cancel out, simplifying the calculation to:

$ \text{Number of balls} = \frac{R^3}{r^3} = \left(\frac{R}{r}\right)^3 $

Final Result

Substitute the given values for $R$ and $r$:

$ \text{Number of balls} = \left(\frac{4}{2}\right)^3 = (2)^3 = 8 $

Thus, 8 balls of radius 2 cm can be made from the solid sphere of radius 4 cm.