#### 80.45
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- So, $ A = 40(1.15)^5 $.
- Calculating: $ 1.15^5 \approx 2.011357 $, so $ A \approx 40 \times 2.011357 = 80.45428 $.
- Rounding to two decimal places, the amplitude is approximately 80.45 units.
- 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.
