Question

a right circular cone is divided by a plane parallel to its base into two equal volumes. The ratio of the heights of the the two cones is ​

Answer 1

Let the radius of the base of the cone =  R

Height of the base of the cone = H

Slant height of the base of the cone = L

Again, let the radius of the base of the cone = r

Height of the base of the cone = h

Slant height of the base of the cone = l

Height of the remaining part of the cone = H - h

Given that the right circular cone is divided by a plane parallel to its base in two equal volumes,

So, the volume of the small cone is half of the volume of the whole cone.

=> πr² h/3 = (1/2)×πR² H/3

=> r² h = R² H/2

=> r² h/R² H = 1/2

=> r² /R² = H/2h --->(1 )

=> r/R = h/H

=> (r/R)² = (h/H)²

=> r² /R² = h² /H² ------->(2)

From equation 1, we get

  H/2h = h2 /H2

=> H/2h = (h/H)2

=> (1/2)×(h/H) = (h/H)2

 =>  1/2 = (h/H)³

=> h/H = (1/2)^1/3

=> h/H = 1/21/3

=> H = h×21/3

Now, h/(H - h) = h/(h×21/3 - h)

=> h/(H - h) = h/{h(21/3 - 1)}

=> h/(H - h) = 1/(21/3 - 1)

=> h : (H - h) = 1 : (21/3 - 1)

So, the ratio of the line segment into which the axis of the cone is divided by the plane is 1 : (21/3 - 1)

HOPE IT HELPS !!