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

• 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 the smaller cone \( C \) to the volume of the frustum \( F \), we need to analyze the information provided by the statements individually.

Concept: The volume of a cone is given by the formula V = \frac{1}{3}\pi r^2 h.

The volume of a frustum of a cone is given by the formula:

V_{\text{frustum}} = \frac{1}{3} \pi h (R^2 + Rr + r^2)

where h is the height of the frustum, R is the radius of the base of the original cone, and r is the radius of the top of the frustum.

1. Using Statement 1: "PQR is twice the radius of C"
   • Let the radius of cone \( C \) be \( r \) and its height be \( h \).
   • Then, the radius of the original cone \( PQR \) is \( 2r \), and its height is \( 2h \) (since the cone is similar and cut parallel at its middle height).
   • The volumes of the cones are proportional to the cube of their linear dimensions. Thus, the volume ratio of cone \( C \) and the original cone \( PQR \) is \( \left(\frac{r}{2r}\right)^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).
   • Thus, the volume of the frustum \( F \) is \( \frac{7}{8} \) of volume of the original cone. Hence, the ratio of the volume of \( C \) to the volume of \( F \) becomes:

   \frac{V_C}{V_F} = \frac{\frac{1}{8} V_{PQR}}{\frac{7}{8} V_{PQR}} = \frac{1}{7}

2. Using Statement 2: "PQR is cut off at the middle of its height"
   • This statement directly provides that the top smaller cone \( C \) and the rest, frustum \( F \), are each half the height of the original cone \( PQR \).
   • As the cones are similar, the volume ratios still come out from the cube of the linear dimensions, resulting in \( \frac{1}{8} \) for \( C \) and \( \frac{7}{8} \) for \( F \), like in Statement 1.
   • Thus, again we find:

   \frac{V_C}{V_F} = \frac{\frac{1}{8} V_{PQR}}{\frac{7}{8} V_{PQR}} = \frac{1}{7}

Conclusion: Either statement (1) alone or statement (2) alone is sufficient to determine the ratio. Thus, the correct answer is: Either statement (1) alone or statement (2) alone is sufficient to answer the question.