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Question: The displacement of a buoy, in metres, in the water is described by the function b(t) = 5sin(30t) at time t seconds. If the buoy starts at the center, when is the second time that the buoy will be back at the center?
The buoy will be back at the center whenever the sine function equals zero. The sine function equals zero at nπ, where n is an integer. Here, we have b(t) = 0 when 30t = nπ. Given that the buoy starts at the center, this means that time t=0 seconds is the first time it's at the center. Solving the equation, we get: 30t = π (for n = 1, which is the next time it will be at the center after the start time) So, t = π / 30 ≈ 0.1047 seconds (which is the second time it will be at the center) Then, we find the next time: 30t = 2π (for n = 2, which is the next time it will be at the center after t = π / 30) So, t = 2π / 30 = π / 15 ≈ 0.2094 seconds. Therefore, the second time that the buoy will be back at the center is approximately at t=0.2094 seconds.
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