# Question 2740: Area of a Half-Open Rectangle with Nonnegative Dimensions

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0004-02740 · q-20261005-advance-20261006-r06-area-0015



## Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,14)×[0,5) has width 14 and height 5, both nonnegative. Find its area in squared length units.


- G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open rectangle [0,14)×[0,5) has width 14 and height 5, both nonnegative. Find its area in squared length units.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-area](https://ocme.evemisslab.com/en/mko/mko-area/)





## Step-by-step derivation

S001 · impliesBefore

orthonormal equal-length coordinate units;half-open regions

After

area in squared length units

Conditions: none

Axes are orthogonal with equal length units;half-open boundaries make adjacent unit cells disjoint.Area uses squared units,not length units.

Step identifiers and dependenciesRule: half_open_area_convention · Uses: G001





S002 · calculationBefore

rectangle area := mul(14, 5)

After

rectangle area = 70

Conditions: none

Multiply nonnegative width by height;adding the side lengths does not give area.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

w=14; h=5

After

area=70; empty=No

Conditions: none

A zero side is an allowed empty region;this is not a perimeter calculation.

Step identifiers and dependenciesRule: rectangle_area_boundary · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Recompute rectangle area with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

70

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: f8b24405d772b19edc06b7774600b6f7df2de15ac0f8bc3e9b65e7b0a95d1a46
- candidate record SHA-256: 9a66c6c2fc0891ce377ff09bd06f1d6cc349c6811fdf5209b724e9d120142b67
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r06-area-0015.json)