64 identical small spheres are cast from a sphere of radius 12 cm, with the total volumeof the small spheres being equal to the volume of the larger sphere. The diameter (incm) of each of the small spheres is:

Correct option is A

Given:

Radius of the larger sphere = 12 cm

Number of smaller spheres = 64

The total volume of the smaller spheres is equal to the volume of the larger sphere.

Formula Used:

The volume of a sphere is given by the formula:

V = (4/3)πr³

Where r is the radius of the sphere.

Solution:

1. Volume of the larger sphere:

​$$V_{\text{large}} = \frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi (1728) = 2304 \pi \, \text{cm}^3$$​​

Volume of one small sphere:

The total volume of the smaller spheres is equal to the volume of the larger sphere, so:

Total volume of 64 small spheres = 2304π

Volume of one small sphere = 2304π / 64 = 36π cm³

3. Radius of the small sphere:

Using the volume formula for the small sphere:

​$$\frac{4}{3} \pi r_{\text{small}}^3 = 36 \pi \\\ \\r_{\text{small}}^3 = \frac{36 \times 3}{4} = 27\\\ \\r_{\text{small}} = \sqrt[3]{27} = 3 \, \text{cm}$$​​

 Diameter of the small sphere:

The diameter is twice the radius:

​$$d_{\text{small}} = 2 \times r_{\text{small}} = 2 \times 3 = 6 \, \text{cm}$$​​