Model a cup of tea
Tonight's idea
Hot tea loses heat to the cooler room. The bigger the difference between them, the faster the heat flows. So tea cools quickly while it is very hot and slowly once it is only warm.
Turning a real question into math(s), working out an answer, then checking it against the real thing is called modelling. Tonight's model watches one number, the gap: the tea's temperature minus the room's. Our room stays at 20 °C.
The model rule: each minute, the tea loses 10 percent of the gap. New gap = old gap times 0.9.
We picked the 10 percent. A real cup has its own number, found by measuring. The rule is a simple form of Newton's law of cooling, and it is an approximation: steam, for one, carries heat away in its own way.
Watch one: the first three minutes
The tea starts at 90 °C in a 20 °C room. The first gap is \(90 - 20 = 70\) degrees. Each minute, multiply the gap by 0.9. Then add the room's 20 back to get the temperature.
| Minute | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Gap | 70 | \(70 \times 0.9 = 63\) | \(63 \times 0.9 = 56.7\) | \(56.7 \times 0.9 = 51.03\) |
| Tea (°C) | 90 | 83 | 76.7 | 71.03 |
The tea drops 7 degrees in the first minute, then 6.3, then 5.67. Each drop is 10 percent of the gap at the start of that minute, so the drops shrink as the gap shrinks. That is "fast at first, then slowly". Careful: it is 10 percent of the gap (7 is 10 percent of 70), not 10 percent of the temperature.
Your turn
- Carry the table on for minutes 4, 5 and 6. Use a calculator, keep all its digits, and round only what you write down, to one decimal place. How far does the tea drop in the 6th minute?
- After how many whole minutes is the model tea first below 60 °C? The picture shows about 44.4 °C after 10 minutes. Carry on from there: when is it first below 40 °C?
- Sam's rule: "each minute the tea loses 10 percent of its temperature". Find Sam's tea after 1 and 2 minutes, from 90 °C. After 15 minutes Sam gets about 18.5 °C. Why is that wrong in a 20 °C room?
- Here are some thermometer readings. They are made-up example data, not a real experiment.
Build a second model that loses 8 percent of the gap each minute (new gap = old gap times 0.92) for minutes 1 to 4. Which model is closer to the readings, 8 percent or 10 percent?Minute 0 1 2 3 4 Reading (°C) 90 84 79 75 70 - List two things about a real cup of tea that the model ignores.
Stretch: how long to halve the gap?
The gap starts at 70. Between which two whole minutes does it fall to half, 35? Would the answer change for tea starting at 60 °C? Between which two minutes does the gap fall to a quarter of 70?
Answers
- Gaps: \(51.03 \times 0.9 = 45.927\), then \(41.3343\), then \(37.20087\). Adding 20 and rounding: 65.9, 61.3 and 57.2 °C. During the 6th minute the tea drops from 61.3343 to 57.20087, which is about 4.1 degrees. Compare that with 7 degrees in the first minute.
- From question 1, the tea is at 61.3 °C after 5 minutes and 57.2 °C after 6, so it is first below 60 after 6 minutes. At 10 minutes the gap is about \(44.4 - 20 = 24.4\). Then \(24.4 \times 0.9 = 21.96\), so about 42.0 °C after 11 minutes, and \(21.96 \times 0.9 = 19.764\), so about 39.8 °C after 12 minutes. It is first below 40 after 12 minutes. The first 6 minutes took off about 33 degrees. The next 6 took off only about 17.
- Sam gets \(90 \times 0.9 = 81\) °C, then \(81 \times 0.9 = 72.9\) °C. (Our model says 83 and 76.7.) Sam's rule never uses the room at all. It would say the same thing in a warm kitchen or a cold shed, and it keeps heading down towards 0 °C. But tea left out settles at about the temperature of the room, because heat flows from the warmer thing to the cooler one and stops when they match. In our model the gap shrinks but stays above zero, so the tea closes in on 20 °C.
- The 8 percent model: gaps \(70 \times 0.92 = 64.4\), then \(59.248\), \(54.50816\), \(50.1475\) (rounded). Temperatures: 84.4, 79.2, 74.5, 70.1. These miss the readings by 0.4, 0.2, 0.5 and 0.1 degrees. The 10 percent model gives 83, 76.7, 71.0 and 65.9 (the last from question 1), which miss by 1, 2.3, 4.0 and 4.1 degrees, and the misses keep growing. 8 percent is closer for these readings. That does not make 8 exactly right. It is the better of the two numbers we tried.
- Many answers are fair. Steam carries heat away, most of all while the tea is very hot. The cup itself has to warm up. The room is not exactly 20 °C everywhere, and the air just above the tea is warmer. Someone may stir it, add milk, drink some, or put a lid on. The tea is not the same temperature all the way through. The model squeezes all of this into one number, which is why you check it against readings.
Stretch. After 6 minutes the gap is 37.2 (question 1), still above 35. After 7 minutes it is \(37.20087 \times 0.9 = 33.48\) (rounded), below 35. So the gap halves between minutes 6 and 7. The starting temperature does not matter: seven minutes multiply any gap by \(0.9^7\), which is about 0.478, a little less than a half, and six minutes multiply it by \(0.9^6\), about 0.531, a little more. Tea starting at 60 °C has a gap of 40, which is 21.3 after 6 minutes and 19.1 after 7, so 20 is passed between minutes 6 and 7 again. For a quarter, 70 must fall to 17.5. After 13 minutes the gap is about 17.8 and after 14 minutes about 16.0, so it happens between minutes 13 and 14, close to two halvings in a row.