Two spheres of metal weigh 1 kg and 7kg . The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.

To find the diameter of the new sphere formed by melting the two metal spheres, we can use the principle of conservation of mass. The total mass before and after melting should be the same.

Let's denote the radius of the larger sphere as 'R' (we need to find this), and the density of the metal is assumed to remain constant.

The volume of a sphere can be calculated using the formula: V = (4/3) * π * r^3, where 'r' is the radius of the sphere.

Before melting, the total volume of the two spheres is the sum of their individual volumes:

Volume of the smaller sphere = (4/3) * π * (3 cm)^3
Volume of the larger sphere = (4/3) * π * R^3

The total volume before melting is given by:

Total volume before melting = (4/3) * π * (3 cm)^3 + (4/3) * π * R^3

The total mass before melting is given as 1 kg + 7 kg = 8 kg.

After melting, the two spheres form a new single sphere with a radius of 'R' and volume given by:

Total volume after melting = (4/3) * π * R^3

According to the principle of conservation of mass, the total volume before melting is equal to the total volume after melting:

(4/3) * π * (3 cm)^3 + (4/3) * π * R^3 = (4/3) * π * R^3

Now, we can solve for 'R':

(4/3) * π * (3 cm)^3 + (4/3) * π * R^3 = (4/3) * π * R^3
(4/3) * π * (27 cm^3) = (4/3) * π * R^3 - (4/3) * π * R^3
(4/3) * π * (27 cm^3) = 0

This equation leads to 0 = 0, which means that the value of 'R' doesn't matter. In other words, the two spheres will melt to form a new sphere with the same radius as the larger sphere.

So, the diameter of the new sphere will be twice the radius of the larger sphere, which is:

Diameter of the new sphere = 2 * R

Since we don't have an explicit value for 'R', we can't provide a numerical answer. The information provided in the question seems insufficient to determine the diameter of the new sphere.