Question

"tidal wave" because its effect is rapid change A tsunami (commonly called in tide) is a fast-moving ocean wave caused by an underwater earthquake. The water first goes down from its normal level, then rises an equal distance above its normal level, and finally return to its normal level. Suppose the normal depth of the water is 9 metres. Consider the graph of this sinusoidal wave given below a 10 22.5 30 37.5 15 7.5 Minutes 20 Metres (a) This graph shows periodic behaviour. Explain what this means by making reference to the graph (b) According to the graph, what will be the minimum depth of the water? How do you interpret this answer in terms of what will happen in the real world? (c) State the graph's maximum, amplitude, and period. (d) Is there any vertical shift in the graph? If yes, find the value of the vertical shift. (e) Using the information obtained in (a)-(d), write the equation of trigono- metric function of the graph.

Answer 1

a) A graph is being periodic means a particular part of the graph is repeated again and again consecutively after a fixed interval. Here we can observe that in the sinusoidal graph, after 15 minutes the same graph is repeated upto 30 minutes. Hence this graph shows periodic behavior. (This period is better explained in part (c)).

b) In the graph, the lowest water level that is attained is -1 metres. Hence minimum depth of the water is -1 meters. Considering normal depth of the water is 9 metres, the lowest is 10 metres below the normal water level or commonly known as sea level. To sum it up, in the real world, the minimum depth of the water will be 10 metres below the sea level (or normal water level)

c) This graph attains its maximum at the water level of 19 metres which can be observed directly from the graph.

The amplitude of the sinusoidal wave is : ( maximum-minimum )/2 = ( 19-(-1) )/2= (19+1)/2 =10 metres.

The period of the tidal wave is 15 minutes as discussed in part (a) because the part of the graph from x=0 minutes to x=15 minutes is repeated exactly, from x=15 minutes to x=30 minutes.

d) Considering the normal water level to be at y=0, we can see that the graph is shifted vertically in the positive direction by 9 units. (This is just the matter of frame of reference.)

e) The given graph can be represented by the trigonometric function sine because sine is periodic and with a proper multiplication factor for angle and the function, we can get the required graph. But value of sine increases after x=0 which is not the case here. So the function should be negative of sine. We want period to be 15 units and amplitude to be 10 units. Based on these, the function will be
10 sin 15
which gives values from -10 metres to 10 metres. We want the same function shifted vertically by 9 metres so that the equation of the given graph is
9 15 10 sin
.

To verify, at x=0 we get y=-10(0)+9=9, at x=3.75 we get y=-10(1)+9=-1, at x=7.5 we get y=-10(0)+9=9 and so on. So the function works for the given graph.