Answer :

Given,

• The volume of the cone.

• The cone is right circular cone.

• The cone has least curved surface.

Let us consider,

• The radius of the circular base be ‘r’ cms.

• The height of the cone be ‘h’ cms.

• The slope of the cone be ‘l’ cms.

Given the Volume of the cone = πr^{2}l

----- (1)

The Surface area cylinder is = πrl

S = πrl

[substituting (1) in the Surface area formula]

[squaring on both sides]

----- (2)

For finding the maximum/ minimum of given function, we can find it by differentiating it with r and then equating it to zero. __This is because if the function Z has a maximum/minimum at a point c then Z’(c) = 0.__

Differentiating the equation (2) with respect to r:

[Since and ]

------- (3)

To find the critical point, we need to equate equation (3) to zero.

---- (4)

__Now to check if this critical point will determine the minimum surface area of the cone, we need to check with second differential which needs to be positive.__

Consider differentiating the equation (3) with r:

[Since and ]

Now let us find the value of

As , so the function Z = S^{2} is minimum

Now consider, the equation (4),

Now substitute the volume of the cone formula in the above equation.

π^{2}r^{4}h^{2} = 2 π^{2}r^{6}

2r^{2} = h^{2}

Hence, the relation between h and r of the cone is proved when S is the minimum.

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