The function `H(t)=2.5cos({2pi(t-3)}/12)+4` represents the oceans tide, when will the tide height be 2.5 m?

H(t) is measured in metres, and t is measured in hours after midnight.

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To find the time when the tide is at 2.5m, we need to set the height H(t) to 2.5 and then solve the resulting equation for time t. Notice that the period of the function is `T={2pi}/{{2pi}/12}=12` hours.

This means we are solving the equation

`5/2cos({2pi(t-3)}/12)+4=2.5` move 4 to other side

`5/2cos({2pi(t-3)}/12)=-3/2` multiply by 2/5

`cos({2pi(t-3)}/12)=-3/5`

Now to solve this equation, we see that cosine is negative in the second and third quadrants. By the CAST rule, the associated angle is

`theta=cos^{-1}(3/5)approx0.9273`

Which means that the solutions to the equation in the domain (0,24) are

`{2pi(t-3)}/12=pi+-theta` simplify left side and divide

`t-3=6+-{6theta}/pi`

`t=9+-{6theta}/pi`

`t approx 7.2, 10.8` hours after midnight

which is approximately at 7:12am and 10:48 am.

**The tide height is at 2.5m at 7:12am and 10:48am.**

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