Question

If the voulme of a sphere and cube are equal are equal. Find the ratio of their total surface area

Answer 1

GIVEN: Volume of a sphere = Volume of cube

TO FIND:

Total surface area of sphere / TSA of cube =?

OR (4Pi r²) / (6a²) =?

Volume of a sphere = (4/3) pi r^3 (r= radius of the sphere)

Volume of a cube = a^3 ( a= side of the cube)

(4/3) pi r^3 = a^3

=> r^3 / a^3 = 3/(4pi)

=> r/a = [3/(4pi)] ^1/3 …………..(1)

(4 pi r²) / (6a²)

= [4pi (3)^1/3 * (3)^1/3] / [6*(4pi)^1/3 * (4pi)^1/3]

= [4pi(9)^1/3] / [6*(4pi*4pi)^1/3 ]

Now by rationalizing the denominator by multiplying by (4pi)^1/3

=[ 4pi *(9)^1/3 *(4pi)^1/3] / [6*4pi]

= [(36pi)^1/3] / 6

Or, [cube root of 36pi] / 6 ……….ANS

Answer 2

Hi there!

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

Given,
Surface area of sphere = Surface area of cube

4πr² = 6a²

(r/a)² = 3 / 2π

r / a = √(3/2π)

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

= (4π/3)(r/a)³

= (4π/3)[√(3/2π)]³

= (4π/3)(3/2π)[√(3/2π)]

= 2√(3/2π)

= √(4x3/2π)

= √(6/π)

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

Cheers!