Question

hollow cone is cut by a plane parallel to the base and the upper portion is removed, if CSA of remained is 8/9 of CSA of the whole cone ,find the ratio of the line segment into which the alitutude of the cone is divided by the plane

Answer 1

Answer:

1 : 2

Step-by-step explanation:

Let r be the radius of the base of the cone and L be its slant height.

The CSA of the full original cone is then π r L.

Let's put

k = ( height of upper portion) / ( height of lower portion ).

By similar triangles with height, radius, slant height, we then have

slant height of upper portion = k × slant height of original cone = k L

and

radius of upper portion = k × radius of original cone = k r.

So the CSA of the upper portion is

π ( k r ) ( k L ) = k² π r L.

Taking this away from the original, the remaining CSA is then

( 1 - k² ) π r L.

As this is 8/9 of the original, we have

1 - k² = 8/9  =>  k² = 1 - 8/9 = 1/9  =>  k = 1/3.

So the part of the original altitude that corresponds to the upper portion is just the upper 1/3 of it.  In other words, the original altitude is divided into the ratio 1 : 2.   [ Note, the '1" refers to the 1/3 of the altitude that is removed, and the "2" refers to the 2/3 of the altitude that remains. ]