 Hello and welcome to this session. In this session we will discuss a question which says that the water depth in a narrow channel varies with the types the following table shows the water level at a plan R period. Now here in the first part we have to make a scatter plot of the water depth data. In the second part we have to find a function that models water depth with respect to time, both sine function and cosine function. And in the fourth part we have to find in one circle between what time the depth is more than or equal to 11 feet. Now before starting the solution of this question you should know how to solve it. And that is the periodic phenomena using trigonometric function is of the type y is equal to a sine of b into x minus c the whole plus k or y is equal to s of b into x minus c the whole plus k. There the equation is equal to absolute value of a is equal to 2 y upon absolute value of b as your k represents vertical shift and c represents field shift or horizontal shift. Now this result will work out as a key idea for solving our given question. Now let us start with the solution of the given question. Now here in this table we have given the water depth in a channel for different times starting from 12 a.m. to 12 p.m. First of all we will need the scatter plot of this data. Now on the graph we will take time on horizontal axis and depth on vertical axis. Now let 0, 1, 2, 3 and so on on horizontal axis represent 12 a.m. 1 a.m. and so on 1, 2, 3, 4 and so on on vertical axis represent the depth in field. Now let us plot the first point on the graph that is 12 a.m. with 9.9 feet. Now we know that 0 on horizontal axis represents 12 a.m. So we will take x is equal to 0 representing 12 a.m. and depth is equal to 9.9 feet. So for x is equal to 0 we will take y is equal to 9.9 and this is the required point with coordinates 0, 9.9. Similarly we will plot all the other points on the graph. So we have plotted all the points on the graph. So this is the started plot for the given data. Now here let the periodic phenomena is given by y is equal to a cos of v into x minus theta whole plus k. Now first of all let us write equation of midline for this periodic function. Now here maximum depth is 11.6 feet and minimum depth is 5.4 feet. Midline is at y is equal to minimum value plus maximum value whole upon 2 which implies y is equal to now here minimum value is 5.4 feet plus maximum value of depth is 11.6 feet. So y is equal to 5.4 plus 11.6 whole upon 2 and this implies y is equal to 17 upon 2 which further gives y is equal to 8.5. Now we know that midline is drawn at y is equal to k 8.5 that means vertical shift is equal to 8.5. Now let us find amplitude. Now amplitude is equal to maximum value minus minimum value whole upon 2. So this is equal to 11.6 minus 5.4 whole upon 2 this is equal to 6.2 upon 2 and this is equal to 3.1. Now from the key idea we know that amplitude is equal to absolute value of A. So absolute value of A is equal to absolute value of 3.1 therefore A is equal to 3.1. Now let us find period. Key idea we know that period is equal to 2 pi upon absolute value of B. Now it shows 12 our period. 20 period approximately the same point that was our period. Therefore 2 pi upon B is equal to 12 which implies B is equal to upon 12 which further gives B is equal to pi by 6 which is approximately equal to 0.52 y is equal to 3.14. Now we have A is equal to 3.1 and B is equal to 0.52 approximately. Now we know that cosine function without phase shift and vertical shift is given by y is equal to A cos of B x. Now putting these values here this will be y is equal to 3.1 of 0.4 drawn a graph and here this green curve is the graph of the periodic function y is equal to 3.1 cos of 0.52 this red curve is the graph of the given data. Now where we have to find the phase shift this green curve is the graph of this periodic function that is without phase shift and vertical shift. We will compare the highest point of green curve and highest point of red curve as we see that the parent curve that is this green curve is shifted and we obtain this red curve. The phase shift is of 2 units to rise. So here C is equal to 2 now here we have C is equal to 2 A is equal to 3.1 B is equal to 0.52 approximately and K is equal to 8.5 of periodic function is given by y is equal to 3.1 cos of 0.5 that is x minus 2 the whole and here K is if the model of periodic function is given by 0.52 into x minus 2 the whole plus 8.5. Now let us state that it is given by y is equal to A sin C the whole plus K as in cosine function the change is only in value of C that is so here also now here we will compare the given graph that is this red curve with the graph of the periodic function y is equal to 3.1 sin of 0.52 that is the graph without phase shift and vertical shift here the maximum that is if we compare the highest points then we see that parent curve that is this green curve is shifted 1 unit to the left and we have obtained this red curve here C is equal to minus 1. So here the equation will be y is equal to x minus of minus 1 will be x plus 1 the whole plus 8.5. Now next we will have to find that in one cycle between what time the depth is more than or equal to 11 feet. Now you can use graph calculators to graph a function that its equation is known. So here let us graph the periodic function y is equal to 3.1 as of 0.52 into x minus 2 the whole plus 8.5 using graph calculator. Now using graph calculator the following graph is obtained. Now from the graph we can see that in one cycle the curve reaches y is equal to 11 twice. Now here you can see that x is equal to 0.8 approximately y is equal to 11 and when x is equal to 3.2 approximately y is equal to 11. Now here as the horizontal axis is representing the time so x is equal to 0.8 can be written as 0.00 hours plus 0.8 hours which is equal to 0.8 into which is equal to 12 plus 48 minutes which is equal to 12.8 am similarly 3.3 12 am thus between 12.3 12 am the depth is more than or equal to 11 feet. So given question and that's all for this session hope you all have enjoyed the session.