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= 4sin (0.48t-0.7) +7,  t>=12 where radians have been used. Select the  two  options which are correct statements about the predictions based on this model. Question 24Select one or more: At midnight the depth is approximately 11 metres.  The model can be used to predict the tide for up to 12 days.  At midday the depth is approximately 3.2 metres. The time between the two high tides is exactly 12 hours.  The smallest depth is 3 metres. The depth of water in the harbour falls after midnight.  The largest depth is 7 metres.

The function given is a sinusoidal function, which is often used to model periodic phenomena such as tides. The general form of a sinusoidal function is `y = A sin(B(x - C)) + D`, where:<br /><br />- `A` is the amplitude, which determines the height of the wave.<br />- `B` determines the period of the wave.<br />- `C` is the phase shift, which determines the horizontal shift of the wave.<br />- `D` is the vertical shift, which determines the vertical position of the wave.<br /><br />In the given function `D = 4sin(0.48t - 0.7) + 7`, we can see that:<br /><br />- The amplitude `A` is 4, which means the height of the wave is 4 metres.<br />- The period `B` is 0.48, which means the wave repeats every `(2π / 0.48)` hours.<br />- The phase shift `C` is 0.7, which means the wave is shifted to the right by `0.7 / 0.48` hours.<br />- The vertical shift `D` is 7, which means the wave is 7 metres above the horizontal axis.<br /><br />Now, let's analyze the statements:<br /><br />1. **At midnight the depth is approximately 11 metres.** To find the depth at midnight, we substitute `t = 0` into the function. This gives `D = 4sin(-0.7) + 7`, which is not approximately 11 metres.<br /><br />2. **The model can be used to predict the tide for up to 12 days.** The period of the wave is `(2π / 0.48)` hours, which is approximately 13.1 hours. This means the wave repeats every 13.1 hours, not every 12 days.<br /><br />3. **At midday the depth is approximately 3.2 metres.** To find the depth at midday, we substitute `t = 12` into the function. This gives `D = 4sin(5.06) + 7`, which is not approximately 3.2 metres.<br /><br />4. **The time between the two high tides is exactly 12 hours.** The period of the wave is approximately 13.1 hours, not 12 hours.<br /><br />5. **The smallest depth is 3 metres.** The smallest depth occurs at the bottom of the wave, which is `D - A = 7 - 4 = 3` metres.<br /><br />6. **The depth of water in the harbour falls after midnight.** The depth of water increases and decreases periodically. It does not continuously fall after midnight.<br /><br />7. **The largest depth is 7 metres.** The largest depth occurs at the top of the wave, which is `D + A = 7 + 4 = 11` metres.<br /><br />So, the correct statements are:<br /><br />- The smallest depth is 3 metres.<br />- The largest depth is 11 metres.