A metallic sphere of radius 12 cm is melted into three smaller spheres. If the radii of two smaller spheres are 6 cm and 8 cm, the radius of the third is?

Given,
Diameter = 2 cm
? r = 1 cm
Now, Total surface area of hemisphere = 3?r²
and curved surface area = 2?r²
Required difference = 3?r² - 2?r² = ?r²
= ? x 1² = ? sq cm

Curved surface area = 2?r²
= 2? x 14 x 14 = 2 x (22/7) x 14 x 14
= 2 x 22 x 2 x 14
= 88 x 14
= 1232 sq cm

Volume of small spheres
= Volume of bigger sphere / Number of small spheres = [(4/3)?(4)³] / 64
= [(4/3) x ? x 4 x 4 x 4] / 64
= 4/3 ? cm³

Let radius of small sphere be r
? 4/3?r³ = 4?/3
? r² = 1 cm
Now, surface area of small sphere = 4?r² = 4? cm²

Let the diameter's of two sphere are d₁ and d₂, respectively.
? Ratio of their surface areas = 4?r₁²/4?r₂²
= (2r₁)²/(2r₂)² = d₁²/d₂²
= (d₁/d₂)² = (3/5)² = 9/25 = 9 : 25

Curved surface area of the sphere = 4?r²
or 616 = 4?r²
? ?r² = 616/4 = 154
? r² = (154 x 7) / 22 = 49
? r = ?49 = 7 cm
? Volume of the sphere = (4/3)?r³
= (4/3) x (22/7) x 7 x 7 x 7
= 4312 / 3 cm³

According to the formula
Percentage increase in surface area = [ 2x + x²/100]%
= [2 x 3 + (3)²/100]%
= [6 + 0.09]%
= 6.09%

According to the formula,
Percentage decrease in surface area = [2 x (-24) + (-24) x (-24)/100]%
= [-48 + 5.76]% = - 42.24%

Here, n = 100%
According to the formula
Percentage increase in volume = [(1 + n/100)³ - 1] x 100%
= [(1 + 100/100)³ - 1] x 100%
= [8 - 1] x 100% = 700%

Volume of solid sphere of radius 4 cm = (4/3)?(4)³ cm³
Volume of hollow sphere = 4/3?[(8)³ - (6)³] cm³
? Weight of 4/3?(4)³ cm³ = 4 kg
= 4(512 - 216)/4³ = 18.5 kg

Volume of prism = Area of base x Height
= (1/2) x 10 x 12 x 20
= 1200 cm²

? Weight of prism = 1200 x 6
= 7200 g
= 7.2 kg