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Home » This is a telescoping series, where most terms cancel out: - Esdistancia

This is a telescoping series, where most terms cancel out: - Esdistancia

📅 March 17, 2026 👤 scraface

Mar 17, 2026

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📌 A theoretical physicist explains that a certain virtual particle fluctuates in energy according to the function $ E(t) = 3\sin(2\pi t) + 4 $, where $ t $ is in nanoseconds. What is the average energy of the particle over one full cycle?

📌 The function $ E(t) = 3\sin(2\pi t) + 4 $ has a sinusoidal component with period $ \frac{2\pi}{2\pi} = 1 $ nanosecond.

📌 The average value of $ \sin(2\pi t) $ over one full cycle is 0, so the average of $ 3\sin(2\pi t) $ is 0.

📌 Thus, the average energy is $ 0 + 4 = 4 $.

📌 A science communicator creates a video showing how bacterial culture grows exponentially. If a culture starts with 500 bacteria and doubles every 3 hours, how many bacteria are present after 12 hours?

📌 The growth follows $ N = N_0 \times 2^{t/T} $, where $ N_0 = 500 $, $ T = 3 $, and $ t = 12 $.

📌 So, $ N = 500 \times 2^{12/3} = 500 \times 2^4 = 500 \times 16 = 8000 $.

📌 There are 8000 bacteria after 12 hours.

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