# The depth ( D metres) of water in a harbour at a time ( t hours)after midnight on a particular day can be modelled by the function D= 3 sin ( 0.49t - 0.6 ) + 6 t less and equal than 13 Where...

The depth ( D metres) of water in a harbour at a time ( t hours)after midnight on a particular day can be modelled by the function

D= 3 sin ( 0.49t - 0.6 ) + 6 t less and equal than 13

Where radians have been used.

Select two options which are true:

-The largest depth is 9 metres

-The smallest depth is 6 metres

-The time between the two high tides is exactly 12 hours

-At midnight the depth is approximately 4.3 metres

-The model can be used to predict the tide for up to 13 days

-The depth of water in the harbour falls after midnight

-At midday the depth is approximately 9 metres

Please explain your answer.

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### 1 Answer

The largest depth is 9 metres.

The time between the two high tides is exactly 12 hours

`D=3sin(.49t-.6)+6 `

`D'=3xx.49cos(.49t-.6)`

D' derivative of D w.r.t D

D' =0 if

cos(.49t-.6)=0

.49t-.6=`pi/2`

t=4.43 rad

`D''=-3xx(.49)^2sin(.49t-.6)`

`D'' }_{t=4.43rad}<0`

`t=4.43 ` gives max depth

D=9 approx.