Sizu Riatto
Watt Happens Next · Chapter 4

Slopes From a Table

A slope is a rate you can read straight off a table. Work it hour by hour and the cooling log shows its secret: the house falls faster when it is warmer than outside, in proportion to the gap.

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Opens on the book's house. Change any number to your own and press Calculate.

A table: times and readings (same order)

The cooling model (optional): the slope is proportional to the gap to outside

Enter the times and the readings, then press Calculate.

The formula

  • Forward difference: (y₂ − y₁) ÷ (t₂ − t₁), the slope from one reading to the next
  • Central difference at a point: (the next − the last) ÷ (their time apart): usually closer to the slope at that point
  • The gap = indoor − outdoor. Newton's law of cooling says the slope = −gap ÷ τ
  • k = 1 ÷ τ; from the table, k ≈ −Σ(slope × gap) ÷ Σ(gap²), a line through the origin
The book's worked example, Chapter 4. The book's night, from 20 °C at 22:00: the first hour falls 0.4 °C, the last 0.3 °C, as the gap to the 4 °C outside shrinks from 16 to 12.5 °C. With τ = 36 hours, k = 1 ÷ 36 = 0.0278 an hour and the model's slope at 22:00 is −16 ÷ 36 = -0.444 °C an hour.
Check it. Plot each hour's slope against that hour's gap: the points should fall near a straight line through zero. If they curve, the outdoor temperature moved.

Honest limits. Differences magnify reading errors: a 0.1 °C misread moves one hour's slope by 0.1 °C an hour. Read the same thermometer, at the same height, every hour. The book's house, its meters and its bill were worked out on paper from published figures, not measured by us: read your own meters and log your own night. Tariffs are illustrative, as seen in November 2026 (shaped like the GB price cap of October to December 2026, with US prices beside), shown without a currency sign; put in your own. No saving is promised. Everything stays in your browser: no sign-up, nothing stored.

These calculators go with Watt Happens Next (Make It Add Up, Book 3), where every number on this page is worked by hand.