Question

A cylinder is placed inside a sphere in such
way that axis of cylinder passes through the
centre of the sphere. The volume of the
sphere in 361. If the radius of the cylinder is
two-third the radius of the sphere, what is
the maximum possible height of the
cylinder? (Volume of the sphere = ror)​

Answer 1

Answer:

Maximum Height of the Cylinder is 2√5 cm

Step-by-step explanation:

Given:- A sphere with radius R and a volume of 36π, a cylinder or radius (2R)/3

To find:- Maximum Height of the Cylinder

Proof:-

Let O be the centre of the sphere through which the axis of the cylinder passes through and let AC be the maximum height of the cylinder and OB & OD be the radius of the sphere

we know that, the volume of the sphere is (4/3)πR³

and here the volume is 36π

Thus,

36π = (4/3)πR³

Cancelling the π's we get

36 = (4/3)R³

R³ = (36 × 3)/4

R³ = 27

R = 3cm

Thus the radius R of the sphere = 3cm

so, OB = OD = 3cm (Radii of the same sphere)

Now,

radius of cylinder = (2R)/3

= (2 × 3)/3 = 6/3 = 2cm

So, AB = CD = 2cm (Radii of the same cylinder)

we know, AB and OB then we can find OA

By Pythagoras theorem

OB² = OA² + AB²

OA² = OB² - AB²

OA² = 3² - 2² = 9 - 4 = 5

Thus,

OA = √5 cm

Similarly,

OC = √5 cm

Now,

Height of cylinder (h) = AC

AC = OA + OC

h = √5 + √5

h = 2√5 cm

Therefore,

the maximum height of the cylinder = 2√5 cm

Hope you understood it........All the best