Question

If the curved surface area of a right circular cylinder inscribed in a sphere of a radius r is maximum then show that height of the cylinder is root 2r

Answer 1

Given:

The curved surface area of a right circular cylinder inscribed in a sphere of a radius r is maximum.

To Show:

Height of the cylinder is √2r .

Solution:

Given the radius of the sphere to be  'r ' .

Let 2h be the height of the cylinder inscribed and R be it's base radius.

Now from the diagram we can observe the right angled triangle with radius of the sphere, height of the cylinder and radius of the cylinder as sides.

In this triangle, x is the angle between height and radius of sphere.

• sin x = R / r
• cos x = h / r

Therefore ,

• R = r sin x
• h = r cos x

Now curved surface area of cylinder = base perimeter x height.

• CSA = 2πR x 2h
• CSA = 2π r sin x *2 r cos x
• CSA = 2πr²sin x cos x = 2πr²sin2x .

Now given that curved surface area is maximum.

Therefore ,

• d(CSA)/dx = 0
• 2πr² 2 . cos2x = 0
• cos 2x = 0
• 2x = π/2
• x = π/4

Therefore Curved surface area is maximum when x = π/4

Therefore , for maximum CSA ,

• h = r cos x = r cos(π/4) = r / √2

Therefore height of the cylinder = 2h = r√2 .

Thus showed that the  curved surface area of a right circular cylinder inscribed in a sphere of a radius r is maximum then height of the cylinder is √2 r .

Answer 2

Answer:

Step-by-step explanation:

Given:

The curved surface area of a right circular cylinder inscribed in a sphere of a radius r is maximum.

To Show:

Height of the cylinder is √2r .

Solution:

Given the radius of the sphere to be  'r ' .

Let 2h be the height of the cylinder inscribed and R be it's base radius.

Now from the diagram we can observe the right angled triangle with radius of the sphere, height of the cylinder and radius of the cylinder as sides.

In this triangle, x is the angle between height and radius of sphere.

sin x = R / r

cos x = h / r

Therefore ,

R = r sin x

h = r cos x

Now curved surface area of cylinder = base perimeter x height.

CSA = 2πR x 2h

CSA = 2π r sin x *2 r cos x

CSA = 2πr²sin x cos x = 2πr²sin2x .

Now given that curved surface area is maximum.

Therefore ,

d(CSA)/dx = 0

2πr² 2 . cos2x = 0

cos 2x = 0

2x = π/2

x = π/4

Therefore Curved surface area is maximum when x = π/4

Therefore , for maximum CSA ,

h = r cos x = r cos(π/4) = r / √2

Therefore height of the cylinder = 2h = r√2 .

Thus showed that the  curved surface area of a right circular cylinder inscribed in a sphere of a radius r is maximum then height of the cylinder is √2 r .