# Exam-Style Question on Circular functions

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 618. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is allowed.

The height of water in metres above Chart Datum* in the Port of Bristol on a day in January is modelled by the function:

$$H(x) = a \sin(b(x-c)) +d$$

where $$x$$ is the number of hours after midnight GMT, and $$a,b,c, \text{ and } d$$ are constants greater than zero.

The following graph shows the height of the water for 19 hours starting at midnight.

The first low tide occurs at 02:23 and next low tide occurs at 14:28.

Throughout the day the height of the water fluctuates between 1m and 13.8m.

(a) Show that $$a = 6.4$$.

(b) Find the value of $$b$$ to two decimal places.

(c) Find the value of $$d$$.

(d) Find the smallest possible value of $$c$$ to one decimal place.

(e) Using the values of $$a,b,c, \text{ and } d$$ found above, find the height of the water at 10:00.

(f) Determine the number of hours, during this January day, for which the tide is higher than 10 metres above Chart Datum.

* Chart Datum is a reference datum or level from which heights and depths on a nautical chart are measured. Chart Datum is determined by the lowest astronomical tide, which is the level of the sea at its lowest possible level. It is also the reference level used to measure the depth of water in a harbour or other body of water.

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