Question

As shown in the adjacent figure, a sphere is placed in a cylinder. It
touches the top, bottom and the curved surface of the cylinder. If radius
of the base of the cylinder is T.
a. what is the ratio of the radii of the sphere and the cylinder?
b. what is the ratio of the curved surface area of the cylinder and the
surface area of the sphere?
c. what is the ratio of the volumes of the cylinder and the sphere?​

Answer 1

a) Ratio of the radii of the sphere and the cylinder = 1 : 1

b) Ratio of the curved surface area of the cylinder and the surface area of the sphere = h : 2T

c) Ratio of the volumes of the cylinder and the sphere = 3h : 4T

• Given,

i) A sphere is placed inside a cylinder touching its top, bottom and curved surface.

ii) Radius of the base of the cylinder = T

a) • Radius of the sphere = Radius of the cylinder (as the sphere touches all surfaces of the cylinder)

=> Radius of the sphere = T

• Therefore, ratio of the radii of the sphere and the cylinder = Radius of the sphere / Radius of the cylinder

=> Ratio of radii = T / T = 1 / 1

Or, ratio= 1 : 1

b) • Curved surface area of the cylinder = 2π × radius × height (h)

• Surface area of the sphere = 4 π × (radius)²

•   Ratio of the curved surface area of the cylinder and the sphere = 2πTh / 4πT² = h / 2T

• Ratio of the Ratio of the curved surface area of the cylinder and the sphere = h : 2T

c) • Volume of the cylinder = π × (radius)² × height (h)

• Volume of the sphere = 4/3 × π × (radius)³

• Ratio of volume of cylinder and sphere = Volume of cylinder / Volume of sphere

=> Ratio = πT²h / (4/3)πT³

Or, ratio = 3h / 4T

Or, ratio = 3h : 4T