According to a certain nation's postal regulation, a package in a rectangular shipping box qualifies for express shipping only if...

Phase 1: Owning the Dataset

Understanding the Constraints

We have a rectangular shipping box with dimensions L, W, and H (in centimeters) where:

• \(\mathrm{L} ≥ \mathrm{W} ≥ \mathrm{H}\) (ordered from largest to smallest)
• \(\mathrm{L} × \mathrm{W} × \mathrm{H} = 18,000\) (volume constraint)
• One dimension equals 20 cm
• Box qualifies for express shipping

For express shipping, the two smallest dimensions must not exceed 20 cm and 40 cm respectively. Since \(\mathrm{L} ≥ \mathrm{W} ≥ \mathrm{H}\), the two smallest dimensions are W and H.

Therefore: \(\mathrm{H} ≤ 20\) and \(\mathrm{W} ≤ 40\)

Visual Representation

Let's create a simple diagram showing our dimension relationships:

L (largest) ≥ W (middle) ≥ H (smallest)
              ↓           ↓
           ≤ 40 cm     ≤ 20 cm
           
L × W × H = 18,000

Phase 2: Understanding the Question

We need to select values for L and W from the choices [10, 18, 20, 30, 36] that could represent the length and width of this box.

Key Insight

Since one dimension is 20 cm and we have the ordering \(\mathrm{L} ≥ \mathrm{W} ≥ \mathrm{H}\), let's determine which dimension equals 20.

Phase 3: Finding the Answer

Case Analysis

Case 1: If H = 20

• Since H = 20, we have \(\mathrm{H} ≤ 20\) ✓ (express shipping requirement satisfied)
• From \(\mathrm{L} × \mathrm{W} × \mathrm{H} = 18,000\): \(\mathrm{L} × \mathrm{W} × 20 = 18,000\)
• Therefore: \(\mathrm{L} × \mathrm{W} = 900\)
• We need: \(\mathrm{L} ≥ \mathrm{W} ≥ 20\) and \(\mathrm{W} ≤ 40\)

Checking combinations from our choices:

• If W = 30, then \(\mathrm{L} = 900 ÷ 30 = 30\)
• Check: Is \(30 ≥ 30 ≥ 20\)? Yes ✓
• Check: Is \(\mathrm{W} = 30 ≤ 40\)? Yes ✓
• This works! L = 30, W = 30

? Stop here - we found our answer.

Quick verification of other cases:

• Case 2 (W = 20): Would require \(\mathrm{L} × \mathrm{H} = 900\) with \(\mathrm{H} ≤ 20\), but no valid combinations exist
• Case 3 (L = 20): Would require \(\mathrm{W} × \mathrm{H} = 900\) with both W, H ≤ 20, which is impossible since \(20 × 20 = 400 < 900\)

Phase 4: Solution

Our analysis shows that H = 20 cm, and the only valid combination from the given choices is:

• L = 30 cm
• W = 30 cm

This satisfies all requirements:

• \(\mathrm{L} ≥ \mathrm{W} ≥ \mathrm{H}\): \(30 ≥ 30 ≥ 20\) ✓
• Volume: \(30 × 30 × 20 = 18,000\) ✓
• Express shipping: \(\mathrm{H} = 20 ≤ 20\) ✓ and \(\mathrm{W} = 30 ≤ 40\) ✓