Where Did I Go Wrong in Solving for the Boat's Entry Time?

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The discussion revolves around solving for the boat's entry time using the tide equation, where the user initially calculates t = 1 but finds the answer key states t = 2. The mistake is identified in the interpretation of the cosine function, specifically in the transition from the equation to the inverse cosine step. Participants suggest that the textbook may contain an error regarding the tidal cycle, indicating that the time between successive low tides should be 12 hours instead of 6. This leads to the conclusion that the textbook answer might be incorrect, as there are typically two tides per day. The conversation emphasizes the importance of verifying the source material in mathematical problems.
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Homework Statement
Please see the picture below
Relevant Equations
Simple harmonic motion
1634178732447.png


Equation for the tide:
$$y=-\cos \left(\frac{2 \pi}{6}t\right)+2$$

When ##y## = 1.5 m :
$$1.5=-\cos \left(\frac{2 \pi}{6}t\right)+2$$
$$-0.5=-\cos \left(\frac{\pi}{3}t\right)$$
$$\frac{\pi}{3}=\frac{\pi}{3}t$$
$$t=1$$

But the answer key is t = 2. Where is my mistake?

Thanks
 
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songoku said:
$$-0.5=-\cos \left(\frac{\pi}{3}t\right)$$
$$\frac{\pi}{3}=\frac{\pi}{3}t$$
How did you get from one line to the other? Show your work.
 
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DaveC426913 said:
How did you get from one line to the other? Show your work.
$$-0.5=-\cos \left(\frac{\pi}{3}t \right)$$
$$\cos^{-1} (0.5)=\frac{\pi}{3}t$$
$$\frac{\pi}{3}=\frac{\pi}{3}t$$
 
Hm.
Ballparking suggests the textbook answer may be wrong.

1634183452163.png


But don't take my word for it; I'm just spit-balling, to jog your thought process.
 
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Thank you very much DaveC426913
 
I agree with your result (and with @DaveC426913). What textbook is this from?
 
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I suspect that the textbook author meant to write: the time between successive low tides is 12 hours (not 6).

After all, there should be about two tides per day (in most places), not four! :-p
 
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Doc Al said:
I suspect that the textbook author meant to write: the time between successive low tides is 12 hours
I think you're right.
 
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Doc Al said:
I agree with your result (and with @DaveC426913). What textbook is this from?
This is question given by teacher

Thank you very much Doc Al and Mark44
 
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