Question:medium

A right circular cone (PQR) is cut into two parts, cone (C) and frustum (F), by a plane parallel to the base. What is the ratio of the volume of C to the volume of F?

Statement 1: PQR has twice the radius of C
Statement 2: PQR has been cut off at the middle of its height

Show Hint

For similar cones, volume scales as the cube of the linear ratio; find that ratio from each statement separately.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is D

Solution and Explanation

Step 1: Assign convenient numbers to the shape.
Let the original cone PQR have base radius $2$ and height $2$, so its volume is proportional to $2^2 \times 2 = 8$ (using $V \propto r^2h$, dropping the constant $\pi/3$ since it cancels in ratios).

Step 2: Apply statement 1 with these numbers.
Statement 1 says PQR's radius is twice that of C, so C has radius $1$.
Since C is similar to PQR, its height is also half of PQR's height, giving C a height of $1$.
Volume of C is proportional to $1^2 \times 1 = 1$.
Volume of F is the rest: $8 - 1 = 7$.
The ratio of C to F is $1:7$, a single fixed number. Statement 1 alone is sufficient.

Step 3: Apply statement 2 with the same numbers.
Statement 2 says the plane cuts PQR exactly at the midpoint of its height, so C's height is $1$, half of PQR's height of $2$.
By similarity, C's radius is also half of PQR's radius, so C's radius is $1$.
This is the identical cone C found in Step 2, giving volume proportional to $1$ and F proportional to $7$.
The ratio is again $1:7$. Statement 2 alone is sufficient.

Step 4: Draw the conclusion.
Both statements, checked completely separately, land on the exact same ratio because a radius-halving and a height-halving describe the same similar cone here.

Final Answer:
Each statement alone is enough to find the ratio $1:7$. \[ \boxed{\text{Option (d): Either statement alone is sufficient}} \]
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