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Answer :
The maximum water depth in the harbor as per the model occurs at 5 and 17 hours within the first 24 hours.Therefore the correct option is a.
The model for the water depth in the harbor presented is f(t) = 4.1 sin (π/6 t - π/3) + 19.7. The water depth reaches a maximum when the sine function inside the model is at its maximum value, which occurs when its argument (π/6 t - π/3) is equal to π/2 or (π/2 + 2πn), where n is an integer. Since the sine function has a period of 12 hours in this model, we expect maxima to occur every 12 hours.
Let's find the first maximum within the first 24 hours. Set the argument of the sine function to π/2:
- π/6 t - π/3 = π/2
- π/6 t = π/2 + π/3
- π/6 t = 5π/6
- t = 5 hours
Thus, the first maximum occurs at 5 hours after t=0. Since the period is 12 hours, the next maximum occurs at 5 + 12 = 17 hours. Within the first 24 hours, the maximum water depths are reached at 5 and 17 hours.
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