Question

A cube and a sphere has equal total surface area find thw ratio of the volume of sphere and cube

Answer 1

Answer:

√(6/π)

Step-by-step explanation:

Hi,

Let the side of the cube be 'a'

Total Surface Area of the cube is given by 6a²

Let the radius of the sphere be 'r'

Total Surface Area of sphere is given by 4πr²

Given that both the cube and a sphere has equal total surface area

=> 6a² = 4πr²

=> a =√ (2πr²/3)

To find ratio of the volume of sphere and cube

= Volume of Sphere/ Volume of Cube

= (4/3πr³)/a³

=4π/3*1/(√2π/3)³

=4π/3*(3√3/2π√2π)

=√(6/π)

Hope, it helped !

Answer 2

Answer: Ratio of the volume of sphere and cube is $\sqrt{ \frac{6}{\pi} }$

Solution:

Total surface area of Cube = $6 {a}^{2}$
here a is side of cube

Total surface area of sphere$= 4\pi {r}^{2}$
here r is radius of sphere

ATQ

$6 {a}^{2} = 4\pi {r}^{2} \\ \\ \frac{ {a}^{2} }{ {r}^{2} } = \frac{4\pi}{6} \\ \\ \frac{a}{r} = \sqrt{ \frac{2\pi}{3} } \\ \\ \frac{r}{a} = \sqrt{ \frac{3}{2\pi} }$

Volume of cube = ${a}^{3} \: {unit}^{3}$

Volume of Sphere$= \frac{4}{3} \pi {r}^{3}$
Ratio=

$= \frac{\frac{4}{3} \pi {r}^{3}}{ {a}^{3} } \\ \\ = \frac{4\pi}{3} ( { \frac{r}{a} })^{3}$
put value of r/a

$= \frac{4\pi}{3} ( \sqrt{ \frac{3}{2\pi} } )^{3} \\ \\ = \frac{4\pi}{3} \times \frac{3}{2\pi} \times \sqrt{ \frac{3}{2\pi} } \\ \\ = 2 \sqrt{ \frac{3}{2\pi} }\\\\= \sqrt{ \frac{12}{2\pi} }\\\\=\sqrt{ \frac{6}{\pi} }$

is the ratio of volume of sphere to volume of cube.