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What is the volume of a certain rectangular solid?
We need to find the volume of a rectangular solid (a box with 6 rectangular faces).
Let's call the dimensions length (l), width (w), and height (h). The volume equals \(\mathrm{l} \times \mathrm{w} \times \mathrm{h}\).
For this value question to be sufficient, we need to determine exactly one possible value for the volume. If multiple volumes are possible, the information is NOT sufficient.
A rectangular solid has exactly 3 distinct face areas. If we know all 3 face areas, we can uniquely determine the volume without finding individual dimensions. This is because of a mathematical relationship: \((\mathrm{l} \times \mathrm{w}) \times (\mathrm{l} \times \mathrm{h}) \times (\mathrm{w} \times \mathrm{h}) = (\mathrm{l} \times \mathrm{w} \times \mathrm{h})^2\).
Statement 1 tells us: Two adjacent faces have areas 15 and 24.
Adjacent faces share exactly one edge, meaning they have one dimension in common. So we know 2 of the 3 face areas are 15 and 24.
We don't know the third face area. This is crucial because without all 3 face areas, we can't determine a unique volume.
Let's see what happens with different values for the third face area:
Scenario 1: Third face area = 10
Scenario 2: Third face area = 40
Since different values for the third face area lead to different volumes (60 ≠ 120), Statement 1 is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: Each of two opposite faces has area 40.
This means one of the three distinct face areas equals 40. But we don't know the other two face areas.
Scenario 1: The three face areas are 40, 10, and 16
Scenario 2: The three face areas are 40, 5, and 50
Since we can have different sets of face areas (all including 40), we get different volumes (80 ≠ 100). Statement 2 is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choices B and D (already eliminated).
Now let's use both statements together.
From Statement 1: Two adjacent faces have areas 15 and 24
From Statement 2: One face area equals 40
Here's the key insight: Since 15 and 24 are adjacent faces (they share an edge), they cannot be opposite to each other. In a rectangular solid, we have exactly 3 distinct face areas.
We already know two of them are 15 and 24 from Statement 1. Statement 2 tells us one face area is 40. Since we can only have 3 distinct face areas total, the third one must be 40!
So the three face areas are: 15, 24, and 40.
With all three face areas known, we can determine the unique volume using our relationship:
Since we get exactly one value for the volume, the combined statements are sufficient.
[STOP - Sufficient!]
This eliminates choice E.
Both statements together provide all three face areas of the rectangular solid, which uniquely determines its volume. Neither statement alone gives us enough information.
Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."