Please help- Math work?

Question

Our teacher set us this question: which would take up less space/ is a better fit: a cube in a sphere or a sphere in a cube (a box in a ball or a ball in a box)?

Answer 1

1) Cube with edge length e in a sphere with radius r :

The largest diagonal of the cube is equal to the diameter of the sphere :

2 r = √3 e (you have to apply two times the Pythagorean theorem to get to the √3)

2) Sphere with radius r in a cube with edge length e :

Then the diameter of the sphere is equal to the edge length :

2 r = e

To measure the fit, we must compare the fraction filled space and that fraction is equal to the filled space divided by the total space.

The volume of a sphere with radius r is (4/3) π r³. The volume of a cube with edge length e is e³.

So in the first case the fraction is :

e³ / ((4/3) π r³)

= e³ / ((4/3) π (√3 e / 2)³)

= 8 (3/4) / (π 3 √3)

= 2 / (√3 π)

= 0.36755

In the second case we have :

(4/3) π r³ / e³

= (4/3) π r³ / (2 r)³

= (1/6) π

= 0.5236

So the latter is a better fit !

Answer 2

Let s be the length of an edge of the cube.

Volume of cube = s³ units³



Case 1: Fit sphere into cube

The largest sphere that can fit in the cube has radius r = s/2.

volume of sphere = (4/3)πr³

= (4/3)π(s/2)³

= (4/3)πs³/8

= ⅙πs³



Wasted space = s³ - ⅙πs³ = (1-⅙π)s³ ≅ 0.476s³

47.6% of a sphere in a cube is wasted space



Case 2: Fit cube into sphere

The length of the diagonal of the cube = s√3.

The smallest sphere that will fit around the cube has diameter of s√3

radius r = s√3/2

volume of sphere = (4/3)πr³

= (4/3)π(s√3/2)³

= (4/3)π·3s³√3/8

= ½πs³



wasted space = ½πs³ - s³ = (½π-1)s³

((½π-1)s³)/(½πs³) = (π-2)/π ≅ 0.363

36.3% of a cube in a sphere is wasted space.

Answer 3

Sphere radius 1 has a volume of 4π/3

Outer cube has a side that is twice the radius of the sphere, so it has a volume of 8

Inner cube of side 2/√3 the radius of the sphere, so it has a volume of 8/(3√3).



Volume inner cube / Volume sphere = 8/(3√3)/(4π/3) = 2/(π√3) ≈ 36.76%

Volume sphere / Volume outer cube = (4π/3)/8 = π/6 ≈ 52.36%



A sphere occupies a greater percentage of a bounding cube, than cube occupies of a bounding sphere.

Answer 4

Hint:

V cube = a^3

V sphere =(4/3)pi r^3



a/ cube in a sphere

cube side 1 ---> sphere's diameter = sqrt3

b/

sphere in a cube

sphere's radius = 1 ---> cube side=2