Santa is testing the sweetness-control dynamics of his magical Hot Cocoa Machine. To observe its behaviour, the elves run a long enchanted Sweetness Testing Strip: the machine will push out a strip of paper at a constant speed, and plot on it the sweetness level that the machine produces, so that the sweetness dynamics get drawn as a shape on the strip, where 1cm width corresponds to the sweetness level of one second.
For the first test, elf Bushy pulls the brown lever at full strength for exactly 1 minute. The machine:
- starts sweetening immediately,
- slowly ramps up,
- peaks right as Bushy lets the lever go,
- then gently ramps back down.
On the strip, this appears as a perfect isosceles triangle, 120 seconds wide and 120 sweetness units tall.
Santa beams: “Ho-ho-ho! A triangle of sweetness! Now let’s make it twice as wide in time and twice as tall in sweetness!”
For the second test, two elves — Bushy and his sister Shinny — pull the lever together at full strength for 2 minutes.
The machine is known to behave in a magical but engineer-friendly way, so to be:
- Linear: two elves → twice the sweetness.
- Time-invariant: doing it later → same shape, just later.
Knowing this, and knowing that Bushy and Shinny are doing such a tes, will Santa get the giant double-wide, double-tall triangle he imagines?
a. 🍫📏 Not quite… He gets something only 120 seconds wide but almost 500 sweetness units tall — the extra sweetness stacks vertically!
b. ☕➡️⬅️ Also no… It becomes almost 500 seconds wide but only 100 sweetness units tall, because the sweetness spreads out over time.
c. 🎄📐 Yes! A perfect 240 seconds × 240 sweetness units isosceles triangle — exactly doubled.
d. ⭐🏔️ No — he gets a trapezoid! The overlapping triangular responses create a flat top.
e. 🍆 I don’t care, I only drink unsweetened aubergine juice.
Related control theory topic: linear, time invariant systems