M3 Further Dynamics - Simple Harmonic Motion HELP!

In a harbour, sea level at low tide is 10m below the level of the sea at high tide. At low tide, the depth of the water in the harbour is 8m. On a particular day, low tide occurs at 1pm and the next high tide occurs at 1:30am. A ship can remain in the harbour safely when the depth of water is at least 12m. The sea level is modelled as rising and falling with simple harmonic motion.

Find the length of time, on this particular day for which it is safe for the ship to remain in the harbour.

I have found the amplitude = 5. and w=2pi/25

However, in the solutions, it uses total time as 12.5 + 2t.

I understand the 12.5 hours is how long high tide lasts for as it will be over 18m for this length of time, but how does the 2t come about?

I would have thought total time would be (12.5 + t) when the tide is 12m and above.

Also, using x=asin(wt), the value of x is taken as 1, am I correct in assuming that x was calculated as being 13-12, where 13 is the middle of the amplitude with 8 being lowest and 18 being highest?

Many Thanks in advance!

What time does t represent?

t represents the time that the water level will be 12m and above

Do you mean the time from when it is 12 m to the time when it is a maximum?
If so, that would account for the multiplication by 2.

Oh ok, so every 12.5 hours, water level fluctuates between 8m and 18m, so at high tide, am I correct in assuming, due to simple harmonic motion, that water is above 12m for 6.25h + t (as 13m is middle of amplitude, and the time for 12m-13m is t), and at low tide, from 8-13m, when it rises again from 8m to 13m it will be 6.25h + t?

so overall its 2(6.25h + t) = 12.5h + 2t?

does that sound correct? Is my thinking of simple harmonic motion accurate when I assume that for exactly half the time (12.5h) the water level rises and falls at a constant rate from 13m-18m, and 8m-13m?

Oh ok, so every 12.5 hours, water level fluctuates between 8m and 18m, so at high tide, am I correct in assuming, due to simple harmonic motion, that water is above 12m for 6.25h + t (as 13m is middle of amplitude, and the time for 12m-13m is t), and at low tide, from 8-13m, when it rises again from 8m to 13m it will be 6.25h + t?

so overall its 2(6.25h + t) = 12.5h + 2t?

does that sound correct? Is my thinking of simple harmonic motion accurate when I assume that for exactly half the time (12.5h) the water level rises and falls at a constant rate from 13m-18m, and 8m-13m?

If t is the time from a height of 12m to 13m then yes, that is correct.
Incidentally though, the water level NEVER rises or falls at a constant rate, but it will take the same time to rise and fall.

ah ok, so can you explain how they worked out total time as 12.5 + 2t then please?

The period of the s.h.m, i.e. the time from the mid-point to one extreme and back to the mid-point again is 12.5. time from 12m to 13m and from 13m back to 12 m is t each time so total time is 12.5+2t.