This is the Important question of class 10 Based on Mensuration Chapter of M.L Aggarwal book for ICSE BOARD.
In this question given that two spheres of the same metal weigh 1 kg and 7 kg. 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 big sphere.
This is the Question Number 09, Exercise 17.5 of M.L Aggarwal.
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Two spheres of the same metal weigh 1 kg and 7 kg. 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 big sphere.
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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.