The correct answer is option (D):
Either statement (1) alone or statement (2) alone is sufficient to answer the question
Let's analyze this geometry problem step-by-step. The goal is to find the ratio of the volume of cone C to the volume of frustum F, created by slicing a larger cone PQR.
Key Concepts:
* Volume of a Cone: (1/3) * pi * r^2 * h, where r is the radius and h is the height.
* Similar Cones: When a cone is cut by a plane parallel to the base, the smaller cone formed (C) and the original cone (PQR) are similar. The ratios of corresponding sides are equal.
* Volume of a Frustum: The volume of a frustum can be found using the formula (1/3) * pi * h * (R^2 + Rr + r^2), where h is the height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base.
Analyzing the Statements:
Statement 1: PQR is twice the radius of C. Let the radius of cone C be 'r' and the height be 'h'. Then the radius of the larger cone PQR is 2r. Since the plane cutting the cone is parallel to the base, the ratio of the heights is also the same as the ratio of radii. The height of the larger cone PQR is therefore 2h.
Volume of C = (1/3) * pi * r^2 * h
Volume of F = Volume of PQR - Volume of C
Volume of PQR = (1/3) * pi * (2r)^2 * (2h) = (1/3) * pi * 8r^2 * h = 8 * Volume of C
Volume of F = 8 * Volume of C - Volume of C = 7 * Volume of C
Ratio of Volume of C to Volume of F = (Volume of C) / (Volume of F) = (Volume of C) / (7 * Volume of C) = 1/7
So, Statement 1 alone is sufficient.
Statement 2: PQR is cut off at the middle of its height. Let 'r' and 'h' be the radius and height of cone C.
Then radius of PQR = 2r and height of PQR = 2h.
Volume of C = (1/3) * pi * r^2 * h
Volume of PQR = (1/3) * pi * (2r)^2 * (2h) = (1/3) * pi * 8r^2 * h = 8 * Volume of C
Volume of F = Volume of PQR - Volume of C
Volume of F = 8 * Volume of C - Volume of C = 7 * Volume of C
Ratio of Volume of C to Volume of F = (Volume of C) / (Volume of F) = (Volume of C) / (7 * Volume of C) = 1/7
So, Statement 2 alone is sufficient.
Conclusion:
Both statements, independently, provide enough information to calculate the ratio of the volumes. Therefore, either statement (1) alone or statement (2) alone is sufficient to answer the question.