Question

A right circular cone (PQR) is cut into two parts, cone (C) and frustum (F) by a plane parallel to base.
What is the ratio of volume of C to the volume of F?
Statement 1: PQR is twice the radius of C
Statement 2: PQR is been cut off at the middle of its height
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is D

To determine the ratio of the volume of cone \(C\) to the volume of frustum \(F\) using the given statements, let's analyze both statements one by one.

Statement 1: PQR is twice the radius of C.

This implies that if the radius of cone \(C\) is \(r\), then the radius of the full cone \(PQR\) is \(2r\). Consequently, as the base radius is halved, the ratio of the volume of cone \(C\) to cone \(PQR\) is \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\) due to similar geometric shapes. Simplifying further, considering the formula for the volume of a cone, the volume \(V\) of a cone is \(\frac{1}{3} \pi r^2 h\). Here, \(h\) will adjust for each cone proportionately.

Statement 2: PQR is cut off at the middle of its height.

This implies that the height of the smaller cone \(C\) is half of the height of the larger cone \(PQR\), leading to a similar geometry proportionally for both radius and height. Therefore, the proportion of volumes again becomes \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\) due to proportional changes in both height and radius for similar cones.

Conclusion: Each statement alone provides sufficient conditions by identifying a geometrical similarity (either by radius in Statement 1 or height in Statement 2) to deduce that the volumes are related by a factor of \(\frac{1}{8}\) for cone \(C\) relative to cone \(PQR\). Thus, the frustum volume is the remaining portion when cone \(C\) is subtracted from cone \(PQR\).

Therefore, the ratio of the volumes, using either statement, is sufficient:
Option: Either statement (1) alone or statement (2) alone is sufficient to answer the question.