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.