The height of a cone is 30 and radius is 10. Find the dimensions h and r of a cylinder created in it with the maximum volume, and show h=3(10-r).

We have a cone of height 30 cm and radius 10 cm. A
cylinder of height h with the radius of the base equal to r is drawn in the
cone.

Now we want to maximize the volume of the cone. The
volume of the cylinder is pi*r^2*h.

To find the optimum
values of h and r we take a two dimensional figure with a rectangle of sides h and 2r in
a triangle with height 30 cm and base 20 cm.

Now equating
the ratios for deriving the tan of the angle we have:

30/10
= h/(10-r)

=> 3(10-r) =
h

=> 30 - 3r = h

The
area of the rectangle is 2r*h

=> 2*r*(30 -
3r)

=> 60r - 6r^2

The
derivative of this with respect to r is 60 - 12r

60 - 12r =
0

=> 12r = 60

=>
r = 5

h = 30 - 15 = 15

For
maximum volume the height is 15 and the radius is 5. And we see that 15 = 3*( 10 - 5) =
30 - 15 = 15.

The required height of the
cylinder is 15 cm and the radius of its base is 5 cm.