Question

the largest possible sphere is kept inside a cube. Then eight identical spheres of maximum possible volume are inserted inside the cube near the corners touching the large sphere and three sides of the cube. What is the ratio of the volumes of the larger sphere and one of the smaller sphere

Answer 1

Solution :-

Let us assume that, the radius of largest possible sphere which kept inside a cube is R cm and radius of 8 smaller sphere of maximum possible volume near the corners is r cm.

we know that,

• Diagonal of cube = √3 * side of cube.
• Radius of sphere = Half of diagonal of cube.

then,

→ R/r = (√3 + 1)/(√3 - 1)

→ R/r = {(√3 + 1)/(√3 - 1)} * {(√3 + 1)/(√3 + 1)}

→ R/r = (√3 + 1)² / {(√3)² - (1)²}

→ R/r = (3 + 1 + 2√3) / (3 - 1)

→ R/r = (4 + 2√3)/2

→ R/r = 2(2 + √3)/2

→ R/r = (2 + √3)

→ R : r = (2 + √3) : 1 .

therefore,

→ Volume of larger sphere : Volume of smaller Sphere = (4/3)π*R³ : (4/3)π*r³ =

→ Volume of larger sphere : Volume of smaller Sphere = R³ : r³

→ Volume of larger sphere : Volume of smaller Sphere = (2 + √3)³ : (1)³

→ Volume of larger sphere : Volume of smaller Sphere = [(2)³ + (√3)³ + 3*2*√3(2+√3)] : 1

→ Volume of larger sphere : Volume of smaller Sphere = [8 + 3√3 + 6√3(2 + √3)] : 1

→ Volume of larger sphere : Volume of smaller Sphere = [8 + 3√3 + 12√3 + 18] : 1

→ Volume of larger sphere : Volume of smaller Sphere = [26 + 15√3] : 1 (Ans.)

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Answer 2

Given : Largest possible sphere is kept inside a cube

Then eight identical spheres of maximum possible volume are inserted inside the cube near the corners touching the large sphere

To Find :  the ratio of the volumes of the larger sphere and one of the smaller sphere

Solution:

Let say Radius of larger sphere = R

and small sphere = r

Large sphere touching all side of cube as its largest

and Small sphere touching 3 sides of cube and sphere

Distance of center of Large Sphere from cube vertex = R√3

Distance of center of Small Sphere from nearest cube vertex = r√3

Distance between Centers of spheres = R + r

=> R√3 = R + r  + r√3

=> R(√3  - 1) = r(√3 + 1)

=> R /r  = (√3 + 1)/(√3  - 1)

=> R/r  =  (√3 + 1)²/2

ratio of the volumes of the larger sphere and one of the smaller sphere

=  (4/3)πR³ /  (4/3)πr³

= (R/r)³

=  ((√3 + 1)²/2)³

= (√3 + 1)⁶/8

= 51.98

≈ 52 times

(√3 + 1)⁶ : 8   or 51.98 : 1

approximately 52 : 1

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