What is the volume of a certain rectangular solid? Two adjacent faces of the solid have areas 15 and 24,...

Understanding the Question

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.

Given Information

• The solid has 6 faces arranged in 3 pairs of opposite faces
• Each pair of opposite faces has the same area
• The three distinct face areas are: \(\mathrm{l} \times \mathrm{w}\), \(\mathrm{l} \times \mathrm{h}\), and \(\mathrm{w} \times \mathrm{h}\)

Key Insight

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: \((\mathrm{l} \times \mathrm{w}) \times (\mathrm{l} \times \mathrm{h}) \times (\mathrm{w} \times \mathrm{h}) = (\mathrm{l} \times \mathrm{w} \times \mathrm{h})^2\).

Analyzing Statement 1

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.

What We Don't Know

We don't know the third face area. This is crucial because without all 3 face areas, we can't determine a unique volume.

Testing Different Scenarios

Let's see what happens with different values for the third face area:

Scenario 1: Third face area = 10

• The three face areas would be: 15, 24, and 10
• Using our formula: \(\sqrt{15 \times 24 \times 10} = \sqrt{3600} = 60\)
• Volume = 60

Scenario 2: Third face area = 40

• The three face areas would be: 15, 24, and 40
• Using our formula: \(\sqrt{15 \times 24 \times 40} = \sqrt{14400} = 120\)
• Volume = 120

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.

Analyzing Statement 2

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.

Testing Different Scenarios

Scenario 1: The three face areas are 40, 10, and 16

• Using our formula: \(\sqrt{40 \times 10 \times 16} = \sqrt{6400} = 80\)
• Volume = 80

Scenario 2: The three face areas are 40, 5, and 50

• Using our formula: \(\sqrt{40 \times 5 \times 50} = \sqrt{10000} = 100\)
• Volume = 100

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).

Combining Statements

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

The Complete Picture

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.

Verifying This Is Sufficient

With all three face areas known, we can determine the unique volume using our relationship:

• \((15) \times (24) \times (40) = 14400\)
• \(\text{Volume}^2 = 14400\)
• Volume = 120

Since we get exactly one value for the volume, the combined statements are sufficient.

[STOP - Sufficient!]

This eliminates choice E.

The Answer: C

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."