Solution:

For same material, density will be same.

Density = mass/Volume

Mass of the smaller sphere, m₁ = 1 kg

Mass of the bigger sphere, m₂ = 7 kg

The spheres are melted to form a new sphere.

So the mass of new sphere, m₃ = 1+7 = 8 kg

Density of smaller sphere = density of new sphere

Let V₁ be volume of smaller sphere and V₃ be volume of bigger sphere.

m₁/V₁ = m₃/V₃

1/V₁ = 8/V₃

V₁/ V₃ = 1/8 …(i)

Given radius of the smaller sphere, r = 3 cm

V₁ = (4/3)r³

V₁ = (4/3) ××3³

V₁ = 36

Let R be radius of new sphere.

V₃ = (4/3)R³

V₁/ V₃ = 36÷(4/3)R³

V₁/ V₃ = 27/R³ …(ii)

From (i) and (ii)

1/8 = 27/R³

R³ = 27×8 = 216

Taking cube root on both sides,

R = 6 cm

So diameter of the new sphere = 2R = 2×6 = 12 cm

Hence diameter of the new sphere is 12 cm.