## anonymous one year ago The depth of water in a tank oscillates every 5 hours. The smallest depth is 0.5, and that the largest depth is 7.5. Find a formula for the depth as a function of time t measured in hours for which the depth at time t=0 is the largest depth.

1. anonymous

Sorry, I have to log off now. Good luck.

2. anonymous

@zepdrix Can you help? :)

3. zepdrix

|dw:1443488320174:dw|So we've got our low and our high point.

4. anonymous

So the amplitude is 7.5-.5/2 which is 3.5?

5. zepdrix

|dw:1443488404092:dw|We start at our highest point. And you've calculated the amplitude correctly, ok good!

6. zepdrix

Where is the middle of our function going to be located?

7. anonymous

Not sure?

8. zepdrix

|dw:1443488549410:dw|

9. anonymous

3.75?

10. zepdrix

3.5 + 0.5 Hmm that doesn't sound right :d

11. anonymous

Ohhhh oops I see now, 4

12. zepdrix

|dw:1443488682271:dw|Ok great. And it completes one full oscillation in 5 hours. So the shape should look something like this, ya?

13. anonymous

Beautiful!

14. zepdrix

So should we model this curve with sine or cosine? What do you think? :) One will be a bit easier than the other. You have to think about your starting point.

15. anonymous

This is where I always have a problem. How exactly do you decide whether to use sine or cosine?

16. zepdrix

|dw:1443488906789:dw|

17. anonymous

Does sine always cross at (0,0)?

18. zepdrix

|dw:1443489123175:dw|So which one seems more appropriate? :)

19. anonymous

cosine!

20. zepdrix

Cosine, ok good.

21. zepdrix

Sine always crosses through the middle point at the start. It won't necessarily be (0,0), especially if we have any kind of vertical shift.

22. anonymous

Ok! Is the period 2pi/5 or did I do that incorrectly?

23. zepdrix

Example: $$\large\rm f(x)=\sin(x)+1$$|dw:1443489226950:dw|Yes, normally this curve would go through (0,0) to start, but it's been shift up by 1.

24. anonymous

Oh ok that makes so much sense

25. zepdrix

The period is 5 hours. The letter b that we're going to stick into our formula:$\large\rm f(x)=A \sin(bx)+d$Is given by $$\large\rm b=\frac{2\pi}{period}$$. So if that's what you meant, then yes. Our b is going to be 2pi/5.

26. anonymous

yep that's what I meant, great!

27. zepdrix

Woops cosine :) not sine, my bad. Let's stick in what we know so far, amplitude and our b,$\large\rm f(x)=3.5 \cos\left(\frac{2\pi}{5}x\right)+d$

28. anonymous

How do we find D?

29. zepdrix

Ummm depends how you like to think of it I guess... I like to think of it as a vertical shift of the middle line.

30. zepdrix

normally that middle line is the x-axis. But now it's up at 4, ya?

31. anonymous

yep

32. anonymous

so is 4 d?

33. zepdrix

sounds right!$\large\rm f(x)=3.5 \cos\left(\frac{2\pi}{5}x\right)+4$ And to be a little more accurate, it's shifting the entire function up by 4. You could calculate that shift from any point on the curve. I think it's just easier to work from the middle.

34. zepdrix

yay team \c:/

35. anonymous

YAY! Thanks so much, you're the bomb!!