Formula Scans

Two-Factor Grid Audit

mass and warming together

The final audit combines the two scans so Q = mc Delta T reads as a summary of the table, not as a memorized rule. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Combine both factors only after scanning them

The material property stays fixed at 5 joules per kilogram per kelvin. The grid varies mass across rows and temperature change across columns.

Q=mcΔTQ = mc\Delta T
Two-factor checkThree representative grid cells are shown on one heat scale.10 Jlow40 Jmiddle90 Jhightwo-factor check

Every cell is a checked multiplication

Each cell multiplies one mass by the same material factor and one temperature change. Moving right changes warming; moving down changes mass.

m\ΔT2 K4 K6 K1 kg10 J20 J30 J2 kg20 J40 J60 J3 kg30 J60 J90 J\begin{array}{c|ccc}m\backslash \Delta T&2\ \text{K}&4\ \text{K}&6\ \text{K}\\1\ \text{kg}&10\ \text{J}&20\ \text{J}&30\ \text{J}\\2\ \text{kg}&20\ \text{J}&40\ \text{J}&60\ \text{J}\\3\ \text{kg}&30\ \text{J}&60\ \text{J}&90\ \text{J}\\\end{array}

Low mass and low warming

The smallest shown heat comes from the smallest mass and the smallest temperature change.

Qlow=1 kg⋅5 J/(kg K)⋅2 K=10 JQ_{\text{low}} = 1\ \text{kg}\cdot 5\ \text{J/(kg K)}\cdot 2\ \text{K} = 10\ \text{J}

Middle mass and middle warming

The middle cell is larger because both direct factors are larger.

Qmiddle=2 kg⋅5 J/(kg K)⋅4 K=40 JQ_{\text{middle}} = 2\ \text{kg}\cdot 5\ \text{J/(kg K)}\cdot 4\ \text{K} = 40\ \text{J}

High mass and high warming

The largest cell uses the largest mass and the largest temperature change, so the heat bar is largest too.

Qhigh=3 kg⋅5 J/(kg K)⋅6 K=90 JQ_{\text{high}} = 3\ \text{kg}\cdot 5\ \text{J/(kg K)}\cdot 6\ \text{K} = 90\ \text{J}

The diagram, grid, and formula agree

The formula is now a summary of visible numeric structure: three mass points, three temperature-change points, and a grid where both factors can change together.

m\ΔT2 K4 K6 K1 kg10 J20 J30 J2 kg20 J40 J60 J3 kg30 J60 J90 J\begin{array}{c|ccc}m\backslash \Delta T&2\ \text{K}&4\ \text{K}&6\ \text{K}\\1\ \text{kg}&10\ \text{J}&20\ \text{J}&30\ \text{J}\\2\ \text{kg}&20\ \text{J}&40\ \text{J}&60\ \text{J}\\3\ \text{kg}&30\ \text{J}&60\ \text{J}&90\ \text{J}\\\end{array}
Two-factor checkThree representative grid cells are shown on one heat scale.10 Jlow40 Jmiddle90 Jhightwo-factor check