# CAT Practice : Mensuration

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From now on, you should not be able to look at a bucket without thinking of it as a cone carved out from another cone.

## Cone and Frustum

Q.1: 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?

Statement (I): H = 20 cm
Statement (II): Volume of small cone: volume of large cone : 1:15

1. If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone.
2. If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.
3. If the question can be answered with either statement alone.
4. If the question cannot be answered with the information provided.

Choice B. If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.

## Detailed Solution

From statement I, 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 (A) and (D).

We are down to choices (A), (B) or (D).

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

i.e. ${1 \over 3}$*π*r2*h : ${1 \over 3}$*π*R2*H is 1 : 15 (r is the base radius of the smaller cone and h is the height of the smaller cone)
or r2 * h : R2 * H is 1 : 15

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

Combining the information in the two statements:

When a section is made the two cones are similar triangles. so ${h \over H}$ = ${r \over R}$
R = ${rH \over h}$

We know H = 20
h = ${{1 \over 15} * {rH \over h} * {rH \over h} * {H \over r}}$ * r

i.e., h3 = ${1 \over 15}$H3. Substituting H = 20, we can get the value for h.
Choice (B) is therefore, the correct answer.

Correct Answer: If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.

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## More questions from Mensuration

Mensuration is the science of measurement. Cylinders, cones, cuboids, rectangular parallelopipeds etc.