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Shocking Average Shoe Size in Men—Size 10.5 or 11? Shop Smarter Today! - Get link 4share

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 2Question: A mathematician studying algebraic topology explores symmetries of a surface modeled by a regular polygon with rotational symmetry of order $ n $. If the polygon has 12 sides, what is the sum of all distinct rotational symmetries (in degrees) that map the polygon onto itself?

📌 Solution: A regular 12-sided polygon (dodecagon) has rotational symmetry of order 12. The full set of rotational symmetries consists of rotations by multiples of $ \frac{360^\circ}{12} = 30^\circ $. These rotations are: $ 0^\circ, 30^\circ, 60^\circ, \dots, 330^\circ $. This forms an arithmetic sequence with first term $ 0^\circ $, common difference $ 30^\circ $, and 12 terms. The sum of an arithmetic sequence is given by:

📌 S = \frac{n}{2}(a + l) = \frac{12}{2}(0^\circ + 330^\circ) = 6 \cdot 330^\circ = 1980^\circ

📌 Thus, the sum of all distinct rotational symmetries is $ \boxed{1980^\circ} $.

📌 Question: A middle school student is building a robot with gears. One gear rotates 48 times per minute, and another rotates 72 times per minute. If both gears start aligned, after how many minutes will they align again?

📌 Solution: To find when the gears align again, we compute the least common multiple (LCM) of their rotation periods. Since they rotate at 48 and 72 rpm (rotations per minute), the time until alignment is the time it takes for each to complete a whole number of rotations such that both return to start simultaneously. This is equivalent to the LCM of the number of rotations per minute in terms of cycle time. First, find the LCM of the rotation counts over time or convert to cycle periods: The time for one rotation is $ \frac{1}{48} $ minutes and $ \frac{1}{72} $ minutes. So we find $ \mathrm{LCM}\left(\frac{1}{48}, \frac{1}{72}\right) = \frac{1}{\mathrm{GCD}(48, 72)} $. Compute $ \mathrm{GCD}(48, 72) $:

📌 Prime factorization: $ 48 = 2^4 \cdot 3 $, $ 72 = 2^3 \cdot 3^2 $, so $ \mathrm{GCD} = 2^3 \cdot 3 = 24 $.

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Also \( 0 + 4y = 2 \times 1 \).
Löse nach \( y \): \( 4y = 2 \) → \( y = \frac{2}{4} = 0.5 \).
### Frage 5
Ein Umweltingenieur modelliert die Konzentration \( C \) eines Schadst...
Stelle die Ungleichung auf: \( 100 e^{-0.05t} < 10 \).
Teile beide Seiten durch 100: \( e^{-0.05t} < 0.1 \).
Nimm den natürlichen Logarithmus: \( -0.05t < \ln(0.1) \).
Da \( \ln(0.1) \approx -2.3026 \), \( -0.05t < -2.3026 \).
Dividiere durch -0.05 (kehre die Ungleichung um): \( t > \frac{2.3026}...
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