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5.5米的棍子能否通过3*4米的门

创建于：2025年5月28日

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问题

5.5米的棍子能否通过3*4米的门

思考

Determining if rod can pass through door

The user asked if a 5.5-meter rod can pass through a 3x4 meter door, likely referring to a diagonal fit through a rectangular opening. To solve this, the longest rod that can pass through a doorway is calculated using a specific formula: ((w^(2/3) + h^(2/3))^(3/2)), based on the "sofa moving problem."

I’m checking if 5.5 meters fits into this calculation with width 3 meters and height 4 meters. Let me think about the math involved.

Reevaluating rod dimensions for door

Okay, now the issue is about rotating the rod inside the plane of the door. The maximum rod length that can pass through is the diagonal of the door, given by sqrt(w^2 + h^2), where w is the width and h is the height. But wait, can you tilt the rod at an angle and make a longer rod fit? This would involve rotating the rod slightly, potentially even with part of it outside. Still, considering the door thickness is negligible, the longest rod you can fit remains the diagonal (w^2 + h^2).

Evaluating rod length through tilted doorway

The user is asking whether a 5.5-meter rod can pass through a 3x4 meter door. The diagonal of the opening is 5 meters, so the rod is too long to pass directly through. However, by tilting the rod at an angle, it could potentially fit within the door's plane. After calculating, the maximum possible length that can fit through the door is about 9.86 meters, which is greater than 5.5 meters. So, yes, the rod should be able to pass through by tilting.

Thought for 1m 37s

回答

可以通过。

理由与计算

把门框想成“横向宽 3 m、纵向高 4 m”两个互成 90° 的“通道”（地面方向和竖直方向）。把棍子水平抬进门后，需要在门框里逐渐抬起，完成一次 90° 转向——这与“搬梯子过 L 形拐角”的经典几何问题完全等价，其可通过的最大长度 L_max 满足

Lmax=(a2/3+b2/3)3/2,L_{\text{max}} =\Bigl(a^{2/3}+b^{2/3}\Bigr)^{3/2},

其中 a 是门宽 3 m，b 是门高 4 m。

  1. 32/3≈2.0803^{2/3}\approx2.080
  2. 42/3≈2.5204^{2/3}\approx2.520
  3. 相加得 4.6004.600
  4. 4.6003/2≈9.86 m4.600^{3/2}\approx9.86\ \text{m}

也就是说，只要棍子长度 <9.86<9.86 米，就可以在门框内一边“斜插”一边“抬头”，顺利转过来。

给定棍子长度为 5.5 米，远小于 9.86 米，自然能够通过这道 3m×4m3 \text{m} \times 4 \text{m} 的门。

直观操作步骤

  1. 先斜插 将棍子一端贴近门框下角，另一端抬高，使棍子与门平面有一定倾角（降低在门平面上的投影长度）。
  2. 外端推进、内端抬高 慢慢把在外的那一端推向门内，同时继续抬高已经进入门里的那一端。
  3. 完成转向 当内端几乎竖直时，整根棍子便已穿过门框。

注意：若门洞厚度很大、或者门内外空间受限，需要留意转动空间，但日常建筑中常见的门框厚度（几厘米到十几厘米）对 5.5 m 细棍的影响可以忽略。

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