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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.A seismologist uses AI to predict aftershocks and models the frequency of secondary tremors as inversely proportional to time after the main quake. If there are 48 aftershocks in the first hour, how many are expected in the third hour (assuming proportional decay per hour)?Since frequency is inversely proportional to time, the number of tremors in hour $ t $ is $ f(t) = \frac{k}{t} $.But over hour 1: $ f(1) = k/1 = 48 $, so $ k = 48 $.