Question

A sphere and a cube have equal surface areas. What is the ratio of the volume of the sphere to that of the cube?

Answer 1

Answer:

The ratio of the volume of the sphere to that of the cube is √6 : √π.

Step-by-step explanation:

Let ‘r’ and ‘a’ be the radius of the sphere and edge of the cube.  

Given :  

Surface area of sphere = Surface area of cube

4πr² = 6a²

r²/a² = 6/4π

(r/a)² = 3/2π

On taking square roots of both sides

r/a = √(3/2π) ......................(1)

Volume of sphere (V1) / Volume of cube(V2) = (4/3)πr³ / a³

V1/V2 = (4/3)πr³ / a³

V1/V2 = (4π/3) (r/a)³

V1/V2 = (4π/3) (√(3/2π))³

[From eq. 1]

V1/V2 = (4π/3) (3/2π)(√(3/2π))

V1/V2 = 2 √(3/2π)

V1/V2 = √(4 x 3/2π)

V1/V2 = √(6/π)

Volume of sphere (V1) : Volume of cube(V2) = √6 : √π

Hence, the ratio of the volume of the sphere to that of the cube is √6 : √π.

HOPE THIS ANSWER WILL HELP YOU ..

Answer 2

✨HOLA✨

Let r and s be the radius of the sphere and edge of the cube respectively.

Given, Surface area of sphere = Surface area of cube.

$4\pi {r}^{2} = 6 {s}^{2}$

Shifting,

$(r \div a) {}^{2} = (6 \div 4)\pi$

$(r \div a) {}^{2} = (3 \div 2)\pi$

$r \div a = \sqrt{(3 \div 2)\pi}$

Now,

volume of Sphere / volume of cube

$=( (4 \div 3)\pi {r}^{2} ) \div ( {a}^{2} )$

$= (4\pi \div 3)( \sqrt{(3 \div 2)\pi} {}^{3}$

= (4π/3)(3/2π)(√(3/2π))

= 2√(3/2π)

= √(4x3/2π)

= √(6/π)

Thus, Volume of sphere : Volume of cube = √6 : √π.

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