Question

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

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

For similar cones, volume scales as the cube of the linear ratio; find that ratio from each statement separately.

Answer 1

The Correct Option is D

Step 1: Set up the similarity relation.
When a cone is cut by a plane parallel to its base, the smaller piece C is a cone similar to the original cone PQR.
If the linear scale factor between C and PQR is k, then the volume of C is \( k^3 \) times the volume of PQR.

Step 2: Test statement 1 alone.
Statement 1 says the radius of PQR is twice the radius of C, so \( k = \frac{\text{radius of C}}{\text{radius of PQR}} = \frac{1}{2} \).
Volume of C \( = k^3 \times \) volume of PQR \( = \frac{1}{8} \) of the whole cone.
Volume of F, the remaining frustum, is \( 1 - \frac{1}{8} = \frac{7}{8} \) of the whole cone.
The ratio of C to F is \( \frac{1/8}{7/8} = \frac{1}{7} \), a fixed number. Statement 1 alone is sufficient.

Step 3: Test statement 2 alone.
Statement 2 says the cut is made at the middle of the height of PQR, so the height of C is half the height of PQR.
Because C and PQR are similar cones, the height ratio equals the linear scale factor, so \( k = \frac{1}{2} \) again.
This gives the same volume split, C is \( \frac{1}{8} \) and F is \( \frac{7}{8} \) of the whole cone, so the ratio is \( \frac{1}{7} \). Statement 2 alone is also sufficient.

Step 4: Compare the two results.
Each statement, used on its own, leads to the same fixed ratio of \( 1:7 \).
Since either statement alone is enough, neither one is required together with the other.

Final Answer:
Both statement 1 alone and statement 2 alone give the ratio \( 1:7 \). \[ \boxed{\text{Option (d): Either statement alone is sufficient}} \]