For the depth of water on Friday, 16 January 1998, the periodic function is:
y = 1.05 sin 2pix/11.72 + 1.55
1.05 is the 'a' value
2pi/11.72 is the 'b' value
1.55 is the 'd' value
we were told not to take into consideration the 'c' value in our equation
'y' is equal to the depth of water
'x' is equal to the number of hours after 6am on Friday, 16 January (6am = 0x, 7am = 1x, 8am = 2x etc.)
the maximum height of water on the graph is 2.6m and the minimum height of the water is 0.5m
With using the equation, the question is asking during what time periods will the water be above 1.7m on Monday, 19 January (3 days later)
I have done this question two ways. I have worked out each individual equation from where x = 66 (12am on 19 January) until when x = 89 (11pm on 19 January)
I have also worked it out by making y = 1.7 and working out the two 'x' values that way. Doing it this way, I ended up with 6:16am and 11:36am.
Which way would be the more right way to do this question?
Took me a while, but I was sure that Steve had answered this for you
http://www.jiskha.com/display.cgi?id=1432887186
and again here
http://www.jiskha.com/display.cgi?id=1432801604
As you figured, 19 January starts 66 hours from x=0. So, you want values of x between 66 and 90 (midnight on Jan 19)
You can see that there are two periods during that time when y >= 1.7
http://www.wolframalpha.com/input/?i=plot+y+%3D1.05+sin%282pi%2F11.72+x%29+%2B+1.55%2C+y%3D1.7+for+66%3C%3Dx%3C%3D90
The actual endpoints of the intervals can be read off here:
http://www.wolframalpha.com/input/?i=solve+1.05+sin%282pi%2F11.72+x%29+%2B+1.55-1.7%3D0+for+66%3C%3Dx%3C%3D90
Both methods you have used to solve the question are correct, but they approach the problem from different perspectives.
The first method you used involves finding the values of 'x' where the depth of water is above 1.7m. This involves solving the equation y = 1.05 sin (2πx/11.72) + 1.55 for y = 1.7. By substituting y = 1.7 into the equation, you can solve for 'x' to find the time periods when the water is above 1.7m. In this case, you obtained 6:16am and 11:36am.
The second method you used involves graphically solving for the values of 'x' where the depth of water crosses the y = 1.7m line. By setting y = 1.7, you can plot this line on the graph of the periodic function y = 1.05 sin (2πx/11.72) + 1.55. The points where the graph intersects the y = 1.7 line correspond to the values of 'x' when the water is above 1.7m. In this case, you obtained the same values of 6:16am and 11:36am.
Both methods are valid and will give you the correct answer. The choice between the two methods depends on your preference and understanding. If you are comfortable with solving equations algebraically, then the first method may be more suitable for you. However, if you prefer graphical methods or find it easier to visualize the problem, then the second method may be more convenient.
Overall, both methods provide a correct approach to solving the question, and it is up to you to decide which method you are more comfortable with.