select the correct answer.
two equal spheres with the maximum possible radius are carved out of a right cylinder.
find the ratio of the volume of one sphere to the volume of the right cylinder.
a.
1 : 1
b.
1 : 3
c.
2 : 3
d.
3 : 1

The ratio of the volume of one sphere to the volume of the right cylinder is 1 : 3 ⇒ answer B

Step-by-step explanation:

* Lets explain how to solve the problem

- The spheres touch the two bases of the cylinder

∴ The height of the cylinder = the diameters of the two spheres

∵ The diameter of the sphere = twice its radius

∴ The diameter of the sphere = 2r, where r is the radius of the sphere

∵ The height of the cylinder = 2 × diameter of the sphere

∴ The height of the cylinder = 2 × 2r = 4r

- The spheres touch the curved surface of the cylinder, that means

  the diameter of the sphere equal the diameter of the cylinder

∴ The maximum possible radius of the sphere is the radius of the

   cylinder

∵ The radius of the sphere is r

∴ The radius of the cylinder is r

- The volume of the cylinder is πr²h and the volume of the sphere is

  4/3 πr³

∵ The height of the cylinder = 4r ⇒ proved up

∵ The radius of the cylinder = r

∵ The volume of the cylinder = πr²h

∴ The volume of the cylinder = πr²(4r) = 4πr³

∵ The radius of the sphere = r

∵ The volume of the sphere = 4/3 πr³

∴ The ratio of the volume of one sphere to the volume of the right

   cylinder = 4/3 πr³ : 4 πr³

- Divide both terms of the ratio by 4 πr³

∴ The ratio = 1/3 : 1

- Multiply both terms of the ratio by 3

∴ The ratio = 1 : 3

∴ The ratio of the volume of one sphere to the volume of the right

  cylinder is 1 : 3