Question

Show that the right circular cone of the least curved surface and given volume has an altitude equal to √2 times the radius of the base.

Answer 1

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 = πr²l

The Surface area cylinder is = πrl

S = πrl

S = πr \((\sqrt{h^2+r^2})\)

[substituting (1) in the Surface area formula]

[squaring on both sides]

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:

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

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:

Now let us find the value of

so the function Z = S² is minimum

Now consider, the equation (4),

9V² = 2π²r⁶

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

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