Question

In the above figure, a sphere is placed in a cylinder. It touches the top, bottom and curved surface of the cylinder. If the radius of the base of the cylinder is ‘r’, write the answer to the following questions.

1. What is the height of the cylinder in terms of ‘r’? 
2. What is the ratio of the curved surface area of the cylinder and the surface area of the sphere? 
3. What is the ratio of volumes of the cylinder and of the sphere? 

Answer 1

a. Given: The cylinder has base radius = r and contains a sphere that touches the top, bottom and the curved surface.

The sphere’s radius = cylinder base radius = r.   ...(Because it touches the curved surface)

The sphere’s diameter = 2r and since it touches top and bottom, the cylinder height = the sphere’s diameter = 2r.

Height of the cylinder = 2r.

b. Given: Cylinder radius = r and height = 2r (From part a).

Curved surface area (CSA) of cylinder = 2πrh

= 2π × r × (2r) 

= 4πr²

Surface area of sphere = 4πR²

= 4πr²

Ratio (CSA of cylinder) : (Surface area of sphere)

= 4πr² : 4πr²

= 1 : 1

The ratio is 1 : 1 they are equal.

c. Given: Cylinder: radius = r, height = 2r; Sphere: radius = r.

Volume of cylinder = πr²h 

= πr² × 2r 

= 2πr³

Volume of sphere = `4/3 πr^3`.

Ratio (cylinder) : (sphere) = `2πr^3 : 4/3 πr^3` 

= `2 : 4/3` 

= 3 : 2

The ratio of volumes (cylinder : sphere) = 3 : 2 (cylinder is `3/2` times the sphere).