the spear and cube have the same surface area show that the ratio of the volume of the sphere to that of the cube is root 6 is to root pie
Answers
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
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