Question

The surface area of sphere and cube are equal ..prove that their volumes are in the ratio 1:root pie/6

Answer 1

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Surface area of sphere = 4πr²

Total surface area of cube = 6a²
(where a is side of cube)
∴ 4πr² = 6a²
r²=6a²/4π
r = √3a²/√2π
r = a√3/√2π
Volume of sphere = 4πr³/3
Volume of cube = a³
Ratio of volumes = 4πr³/3 ÷ a³
=4π(a√3/√2π)³/3 ÷a³ (by putting r=a√3/√2π)
=(4π/3) a³ × (3/2π) × (√3/√2π) ÷ a³
= 2√3/√2π
= √2 × √3/√π = √6/√π
= 1 ÷ √π/√6
= 1 : √(π/6)
Therefore ratio of volume of sphere to volume of cube comes out to be
1 : √(π/6)

PrOvEd

Answer 2

Answer:

Step-by-step explanation:

Given---> Sphere and cube have the same area

To show---> Ratio of volume of sphere and volume of cube

Solution---> We know that ,

1) Surface area of sphere = 4 π r²

2) Surface area of cube = 6 a²

3) Volume of sphere = ( 4/3 ) π r³

4) Volume of cube = a³

Let radius of sphere and side of cube be r and x respectively.

ATQ,

Surface area of sphere = Surface area of cube

=> 4 π r² = 6 x²

=> r² / x² = 6 / 4π

=> r² / x² = 3 / 2π

=> r / x = √( 3 / 2π )

Volume of sphere( V₁ ) = ( 4 / 3 ) π r³

Volume of cube ( V₂ ) = x³

V₁ / V₂ = ( 4 / 3 ) π r³ / x³

= ( 4π / 3 ) ( r³ / x³ )

= ( 4π / 3 ) ( r / x )³

= ( 4π / 3 ) { √(3 / 2π ) }³

= ( 4π / 3 ) {√( 3/2π ) }² √( 3/2π )

= ( 4π / 3 ) ( 3 / 2π ) √(3/2π)

= 2 √(3/2π)

= √( 4 × 3 / 2 π )

= √( 2 × 3 / π )

V₁ / V₂ = √6 / √π

=> V₁ : V₂ = √6 : √π