Sphere volume problem need math help?

Find the radius of the sphere.

sin(45°) = (5.848/2) / r

r = 4.135

Volume of sphere = (4/3)π(4.135)³ = 296.2 m³.

Find the distance from one corner of the cube to the opposite corner. This will give you the diameter of the sphere

s^3 = 200

s = 200^(1/3)

d^2 = s^2 + s^2 + s^2

d^2 = 3s^2

d^2 = 3 * 200^(2/3)

d = sqrt(3) * 200^(1/3)

Now, the volume of a sphere is (4/3) * pi * r^3

r = d/2

V = (4/3) * pi * (d/2)^3

V = (4/3) * (1/8) * pi * d^3

V = (1/2) * (1/3) * pi * (3^(1/2) * 200^(1/3))^3

V = (1/6) * pi * 3^(3/2) * 200

V = (3 * 200 / 6) * pi * sqrt(3)

V = (600 / 6) * pi * sqrt(3)

V = 100 * sqrt(3) * pi

More precisely, the length of each side, s, of the cube is cbrt(200). No need to evaluate it yet.

The length of the diagonal of a cube is given by

d^2 = 3*s^2

so the radius of the sphere is

r = d/2 = (sqrt(3s^2))/2

s = cbrt(200)

so volume of the sphere is

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

= (4pi*((sqrt(3s^2))/2)^3/2

= (4pi*((sqrt(3*(cbrt(200))^2))/2)^3)/3

= 544 m^3