What Height Above the Base is the Section Made? If

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27th of the value of the given cone, at what height above the base is the section made?

1. 19 cm
2. 20 cm
3. 12 cm
4. 15 cm Harsh Vardhan Professor Asked on 16th July 2015 in

Answer: (2) 20 cm
Explanation:-
Let the height and radius of big cone be h1 and r1.
and the height and radius of cut off small cone be h and r.
We know that, from similar triangle,

h1 / r1 = h2 / r2

h2 = h1 r2 / r1

Given-  h1 = 30

h2 = 30 (r2 /r1) ——–(1)

Volume of big cone, V1 = (1/3) π (r12) h1

volume of small cone,V2= (1/3) π r2 h2

V1 / V2 = (r12) h1/ (r22 h = 27 (Given- the volume of small cone is 1/27 of big cone)

(r) h1 = 27(r22) h2

30 (r12) = 27(r22)h2

h1 = 30(r12) / 27 (r22)

h2 = (30 /27)(r1 /r2)2 ———(2)

From equating (1) and (2)

30 (r2 /r1) = (30 /27) (r1/r2)2

(r2 /r1)3 = 1 / 27

r2 / r1 = 1/3

substituting this in (1) h2 = 30 (r2 /r1)

h2 = 30 (1/3) = 10 cm

the height of section made is 30 – 10 = 20 cm.
Hence, the answer is (2) 20 cm.