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UAT-UK SPEC: M3MODULE: MATHEMATICS 1TOPIC ID: ARCH-M1-06

2D/3D Geometric Similarity Scaling (k, k², k³) & Frustums

Master 2D/3D Geometric Similarity Scaling (k, k², k³) & Frustums on the ESAT Mathematics 1 paper. Traditional classroom derivations waste valuable seconds. Using the 2-step Sub-90 invariant elimination method compresses solve time to 35s, saving +110s per question.

Standard Classroom Time
145s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
35s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+110s
Banks clock buffer across 27 questions

2-Step Sub-90 Shortcut Formula

How to eliminate intermediate algebra and solve under 45 seconds.

Launch Interactive Lesson
STEP 1 • SETUP INVARIANT (LINE 1)10–15s
Line 1 (Cube the Ratio): k=43  ⟹  k3=6427k = \frac{4}{3} \implies k^3 = \frac{64}{27}.

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Cancel 2727): 108×6427=4×64=256 cm3108 \times \frac{64}{27} = 4 \times 64 = 256\text{ cm}^3 (20s).

Evaluate the simplified expression using integer factoring or unit elimination.

Distractor Trap Forensics

The predictable cognitive traps engineered by examiners into the wrong options.

TRAP #1factor_of_two_or_stoichiometry_trap

Scaling volume by k² or k instead of k³ (e.g. 192 * (3/4) = 144 instead of 192 * (3/4)³ = 81)

Applying linear (k) or area (k²) scale factor to a 3D volume.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #2subsystem_isolation_trap

Using frustum height ratio h(bottom) / h(total) instead of full similar cone ratio h(top) / h(total)

Applying similarity scaling to the non-similar frustum slice instead of the whole apex cone.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #3intermediate_stop_trap

Outputting the top small cone volume V(small) instead of frustum volume V(large) - V(small)

Forgetting to subtract the removed apex cone from the full cone.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: 2D/3D Geometric Similarity Scaling (k, k², k³) & Frustums

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Full Lesson All 15 Questions (ARCH-M1-06)
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ARCH-M1-06: 2D/3D Geometric Similarity Scaling (k, k², k³) & Frustums (Sample)Mathematics 1
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SUB-90 ZONE (<50S)
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Question 1 of 1ARCH-M1-06 (6-Card Lesson)Format: Standard 5-Option
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Question 1 of 1 • 90s Target

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