Sizu Riatto
Calculus · Chapter 23

Half-Life and Cooling

Decay shrinks at a rate proportional to what’s left, so it halves in equal times. Cooling is the same idea applied to the gap between a drink and the room.

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Opens on Chapter 23’s example: 80 grams with a 5-day half-life; a sample with 60% of its carbon-14 left; Hollis’s coffee at 180°F that reads 160°F after 5 minutes in a 70°F room.

1. What’s left
2. An age from the fraction left
3. Cooling
4. Milk now, or milk later?
Enter your numbers, then press Work it out.

How it works

Something with a half-life H keeps 2^(−t/H) of itself after time t: halve it once for each half-life. The decay constant is k = ln 2 ÷ H, so the amount is A × e^(−kt), and the age of something with a fraction f left is ln(1/f) ÷ k. Newton's law of cooling, a good approximation for a cup of coffee, says the gap between the drink and the room shrinks the same way: T = room + (start − room) × e^(−kt). One reading pins down k, and then the curve tells you any time or temperature. It cools fastest when it's hottest.

From the book, Chapter 23. 80 grams halves to 10 in 15 days, and is 30.3 after 7. With 60% of its carbon-14 left, a sample is about 4,201 years old. Hollis’s coffee: k ≈ 0.0401 a minute, so it reaches 140°F after about 11.3 minutes, cooling at 4.41°F a minute at the start. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.