"Nova only logged every six hours," Comet says. "What was it at hour 2? Hour 14?"
"The model can tell us," Wren says. "T(2) = 6 sin(π/6) + 20. Sine of thirty degrees is a half."
"So 6 times a half plus 20. 23 degrees," Comet says. "Can we check it?"
Nova pulls a second log, the one with the readings between the main hours. "Would you like a hint? Look at hour 2."
"23," Wren reads. "The model fits between the readings too. What do you notice at hour 14?"
"17. Below the midline, on the way down," Comet says. "The wave is falling from hour 6 to hour 18."
"Rising, falling, highest, lowest," Wren says. "Key features. Let us read them all off the wave."
The model T(t) = 6 sin((π/12)t) + 20 gives a temperature for any hour, not only the logged ones.
To use it, multiply the hour by π/12, take the sine, multiply by 6 and add 20. Use the exact sines from week 9.
| Hour | (π/12) × hour | Sine | Model T(t) | Nova's between-hours log |
|---|---|---|---|---|
| 2 | π/6 | 1/2 | 23 | 23 |
| 10 | 5π/6 | 1/2 | 23 | 23 |
| 14 | 7π/6 | -1/2 | 17 | 17 |
| 22 | 11π/6 | -1/2 | 17 | 17 |
Every value in the last column is the crew's own made-up reading. The model matches each one, so the crew keeps it.
Anything that cycles can be modeled this way once you have its high, low and repeat. A swing, a turning wheel, the light on a windowsill.
Each would need its own three numbers from its own readings. The crew's numbers belong to the Glass House only.
| Statement | True or false? |
|---|---|
| The wave is above its midline from hour 0 to hour 12. | ? |
| 6 × 1/2 + 20 = 23 | ? |
| T(6) and T(30) are different values. | ? |
| The y-intercept of the temperature wave is its highest value. | ? |
A full week of waves. Tomorrow is Garden Day: show someone at home how a turning hand makes a wave.