Modeling Tidal Functions: Finding the Best Fit Sine or Cosine Equation

[kateman]

Homework Statement

i was reading this book for school called investigating change and i was really lost on question 4:

tides of an area and their times are as follows:
high tide: low tide
1.64m 12:08am 0.4m 6:03 am
2.32m 12:46 PM 0.57m 7:33PM
1.59m 1:06am 0.49m 6:59am
2.26m 1:41PM 0.56m 8:33PM

a) find the average levels of high and low tides and average time between high tides. use this data to find a sine or cosine function to fit the data as well as possible. Note: the independent variable needs to be a real number, not a number of hours or minutes.

b) graph the data from the table and the function you have found on the same axes.
c)calculate the differences between the data and your model and try to find a function that fits these differences fairly well.

Homework Equations

y = A cos(Bx + C) + D

The Attempt at a Solution

average high tide: 1.9525m
low tide: 0.505m
time between high tide: 12 hours and 31 minutes

easy enough. what i have trouble with is the functions. iam guessing that the function will be a cos function since tides have to start off at a point where as sine starts from 0.

amplitude iam guessing will be: (ave high tide - ave low tide)/2 = 0.72375
and from that the mean value (in other words d) will be: (amplitude + avg low tide)= 1.22875

but i can't determine the period (b) or the phase shift (c)

also, iam lost to the point that i don't know how to make the equation for part a without a time variable. and have i started the part c question first with this equation planning or part a equation? is the time variable the only difference between the two?
i'll appreciate any help, thanks :)

 

[HallsofIvy]

Homework Statement

i was reading this book for school called investigating change and i was really lost on question 4:

tides of an area and their times are as follows:
high tide: low tide
1.64m 12:08am 0.4m 6:03 am
2.32m 12:46 PM 0.57m 7:33PM
1.59m 1:06am 0.49m 6:59am
2.26m 1:41PM 0.56m 8:33PM

a) find the average levels of high and low tides and average time between high tides. use this data to find a sine or cosine function to fit the data as well as possible. Note: the independent variable needs to be a real number, not a number of hours or minutes.

b) graph the data from the table and the function you have found on the same axes.
c)calculate the differences between the data and your model and try to find a function that fits these differences fairly well.

Homework Equations

y = A cos(Bx + C) + D

The Attempt at a Solution

average high tide: 1.9525m
low tide: 0.505m
time between high tide: 12 hours and 31 minutes

easy enough. what i have trouble with is the functions. iam guessing that the function will be a cos function since tides have to start off at a point where as sine starts from 0.

Why do you say that? Doesn't it depend on what time x= 0 corresponds to?

amplitude iam guessing will be: (ave high tide - ave low tide)/2 = 0.72375
and from that the mean value (in other words d) will be: (amplitude + avg low tide)= 1.22875

but i can't determine the period (b) or the phase shift (c)

Doesn't the period depend upon the time between two high tides? And the phase shift depends on the time you choose to correspond to x= 0.

also, iam lost to the point that i don't know how to make the equation for part a without a time variable. and have i started the part c question first with this equation planning or part a equation? is the time variable the only difference between the two?
i'll appreciate any help, thanks :)

Your variable x corresponds to time but you will have to decide: 1) what time corresponds to x= 0 and 2) what time difference is the period of this function?

 

[kateman]

tides arnt ever going to reach 0 with the data we are give
hence x=0 never happens

cos starts off at a point in which tide has a value. sin starts at 0 and since we arnt given a tidal value of 0 it can't be done according to the data


yes, the period would be the time from one high tide to the other. so it would use the average high tide time.

thank-you for helping but iam still confused as to how to do the function in part a. any ideas?

 

[HallsofIvy]

"x" is the independent variable. It refers to when the tides are low or high, not the height, that's y. x can be 0, y can't be.

 

[kateman]

okay, i get that but by the word "when" you mean time which can't be used in the equation.

i don't mean to sound rude but i don't understand what your getting at

 

[HallsofIvy]

Then you had better ask your teacher about this because you simply don't understand what is being asked here. The problem does NOT say "time can't be used in the equation". Since the only things you know are height and time- and you need two variables, one of them must be related to time. What the problem says is that the independent variable, x, won't be the time. It doesn't say it cannot be related to or based on the time. You are, after all, trying to model the height of the tide at any given time.

 

[kateman]

wow, my bad, i was reading the question wrong. thank you for pointing that out to me.

how should i go about relating time to x?
thanks for your patience :)

 

[kateman]

well i did a bit of searching for this function and found that period = 2pie/b and since period would be the time between two high tides that would give me b

phase shift = c/b and since i now have b all i need to know is how to find c.

i really need you to explain this to me because iam on holidays and i don't have any way to ask my teacher for another week. i just don't want to be left behind, please help me.