CAT Data Sufficiency : Mensuration

Here is another Data sufficiency question. The topic is Mensuration.

Question
A right circular cone has height H and radius R. A small cone is cut off at the top by a plane parallel to the base. At what height above the base the section has been made?

A. H = 20 cm
B. Volume of small cone: volume of large cone :: 1:15

Correct answer is Choice C. Both statements are required to answer the question

Explanatory Answer

From statement A, we know that the height of the initial cone is 20cm. However, nothing is said about the small cone. Hence, we cannot answer the question using statement A. So, we can eliminate choices (1) and (4).

We are down to choices (2), (3) or (5).

From Statement B, we know that the ratio of the volume of the small cone to that of the large cone is 1 : 15.

i.e. (1/3)*pi*r^2*h : (1/3)*pi*R^2*H is 1 : 15 (r is the base radius of the smaller cone and h is the height of the smaller cone)

or r^2 * h : R^2 * H is 1 : 15

From this information, we will not be able to find the answer to h. Hence, we can eliminate choice (2).

Combining the information in the two statements:

When a section is made the two cones are similar triangles. so h/H = r/R
R = rH/h

We know H = 20

h = (1/15) ( rH/h) ( rH/h)*H /r*r)

i.e., h^3 = (1/15)H^3. Substituting H = 20, we can get the value for h.
Choice (3) is therefore, the correct answer.