CAT Quantitative Aptitude Questions | CAT Geometry - Mensuration Problems

The question is from CAT Mensuration in CAT Geometry. Mensuration Questions in the CAT exam can be solved with practical approach and thought process rather than just knowing theorems from CAT Geometry. Even if one is not completely prepared for CAT Geometry exam, he or she should be able to do well in Mensuration problems tested by the CAT Exam. Knowing Mensuration formulas is important. Make sure you master Mensuration questions for CAT exam. 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.

Question 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.

Explanatory Answer

Method of solving this CAT Question from Geometry - Mensuration: Let us go Statement by Statement.

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 (C).
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. \\frac{1}{3}\\)*π*r²*h : \\frac{1}{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, \\frac{h}{H}\\) = \\frac{r}{R}\\)
R = \\frac{rH}{h}\\)
We know H = 20
h = \\frac{1}{15}\\) * \\frac{rH}{h}\\) * \\frac{rH}{h}\\) * \\frac{H}{r}\\) * r
i.e., h³ = \\frac{1}{15}\\) * H³. Substituting H = 20, we can get the value for h.
Choice (B) is therefore, the correct answer.

Hence, the answer is 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.

Choice B is the correct answer.