CAT Challenge Qn -1. Mensuration Solution

The solution for the question on mensuration is given in the slideshow below. This is a fairly challenging question on mensuration. The question involves breaking down the given region into simpler geometric shapes and then finding the overlapping area.

Challenge 1 Qn. Mensuration Solution

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CAT Challenging Question Mensuration

Here is an interesting question on mensuration

Question

Two horses are tied to opposite vertices of a rectangular field of length and breadth 6 and 6√2 meters. If they are tied with rope of length 6 meters, what is the common area that both horses can graze?
1. 6 π - 9√3
2. 12 π - 18√3
3. π - 2√3
4. 16 π -8√3
5. Cannot be calculated

People think that questions in 3-d (Cubes, Cuboid, cylinder, sphere etc) are more difficult in mensuration. In my view, the best questions are still from the 2-dimensional shapes. In 3-d, once you know the formula, you can pretty much solve everything. In 2-d, questions can include geometry and mensuration. Those are the best ones.

Anyways, happy solving. Will post solutions soon enough.


Cheers,
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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.