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Home » A_\textbase = \pi (3)^2 = 9\pi \, \textsquare meters - Ready Digital AB

A_\textbase = \pi (3)^2 = 9\pi \, \textsquare meters - Ready Digital AB

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 A retired engineer designs a gear system for a clock where the minute hand gear has 72 teeth and meshes with the hour hand gear. If the minute hand completes 720 rotations in a simulated 24-hour display, how many rotations does the hour hand gear make, given it must turn proportionally to the minute hand?

📌 The ratio of teeth is 72:؟ (unknown, but gears turn inversely to number of teeth).

📌 Since rotations are inversely proportional to number of teeth:

📌 Rotation ratio = 72 / T_hour, but without T_hour, we use motion relationship:

📌 Least common multiple or motion logic: for every 1 rotation of minute gear, its linear speed equals hour gear’s, but gears rotate inversely.

📌 Actually, in a clock: minute gear spins 720 times in 720 minutes, but in a 24-hour display simulated via gear motion:

📌 Standard: minute hand rotates 720 times in 24 hours (1 full rotation every hour, but scaled). Wait — correct: standard minute hand completes 720 rotations in 24 hours? No — 1 rotation per hour → 24 rotations in 24 hours. Contradiction.

📌 Clarify: In 12 hours, minute hand completes 12 rotations. So in 24 hours: 24 rotations. But question says 720 — clearly inconsistent. Re-analyze:

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x + y = 1, \quad x + z = 1 \Rightarrow y = z
From the first equation:
x^2 + y^2 + z^2 = 2 \Rightarrow x^2 + 2y^2 = 2
Let’s try integer values. Try $x = 0$:
+ 2y^2 = 2 \Rightarrow y^2 = 1 \Rightarrow y = \pm1 \Rightarrow z = \p...
So possible coordinates: $(0,1,1)$ or $(0,-1,-1)$.
Check distances:
From $(0,1,1)$ to $(0,0,0)$: $\sqrt{0 + 1 + 1} = \sqrt{2}$
To $(1,1,0)$: $\sqrt{1 + 0 + 1} = \sqrt{2}$
To $(1,0,1)$: $\sqrt{1 + 1 + 0} = \sqrt{2}$

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