Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 7.

The height of the right circular cylinder is 14/√3 and its radius is 7√6/3.

Step-by-step explanation:

Check the attachment for the diagram.

The volume of a right circular cylinder =  πr²h

r is the radius of the cylinder

h is the height of the cylinder

V = πr²h... 1

From the diagram, we can use Pythagoras theorem on the right angled triangle to get r². From the triangle P² = r²+(h/2)²

P is the radius of the sphere.

P = 7

7² = r²+(h/2)²

r² = 49-(h/2)²... 2

Substituting equation 2 into 1:

V = π(49-(h/2)²)h

V = π(49h-h³/4)...3

To get the height h of the cylinder, we need to differentiate the volume of the cylinder with respect to h and equate to zero. i.e dV/dh = 0 since it's the maximum volume.

dV/dh = π(49-3h²/4)

0 = π(49-3h²/4)

Dividing both sides by π

0/π = π(49-3h²/4)/π

49-3h²/4 = 0

49 = 3h²/4

3h² = 49×4

3h² = 196

h² = 196/3

h = √196/√3

h = 14/√3

To get the radius of the cylinder, we will substitute h = 14/√3 into equation 2

r² = 49-(h/2)²

r² = 49-{(14/√3)/2}²

r² = 49-{14/2√3}²

r² = 49-(7/√3)²

r² = 49 - 49/3

r² = (147-49)/3

r² = 98/3

r = √98/√3

r = √49× √2/√3

r = 7√2/√3

r = 7√2/√3 × √3/√3

r = 7√6/3

The height of the right circular cylinders is 14/√3 and its radius is 7√6/3.

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