Among all right circular cones with a slant height of 27, what are the dimensions (radius and height) that maximize the volume of the cone? the slant height of a cone is the distance from the outer edge of the base to the vertex.

Radius = 22.04 unit,

Height = 15.58 unit

Step-by-step explanation:

Since, the volume of cone,

Where,

x = radius,

h = height,

Since, the slant height of a cone is,

Here, l = 27.

So, the volume would be,

Differentiating with respect to x,

For maxima or minima,

But side can not be negative,

So, x = 22.04,

Since,

While,

Thus, by the first derivative test,

V(x) is maximum at x = 22.04,

Also, from equation (1),

h = 15.58

Hence, for maximising the volume, the radius and height of the cone would be 22.04 unit and 15.58 unit respectively.