Posted by grant on Saturday, May 12, 2007 at 9:20am.
The table shows the depth (d metres) of water in a harbour at certain times (t hours) after midnight on a particular day.
time t (hours)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
and the in the next column corresponding to the figures above is
depth d (m)
3.0
3.3
4.2
5.6
7.2
8.2
9.0
8.9
8.1
7.3
5.6
4.3
3.5
3.1
Use the regression facilities on your calculator to fit a sine curve to these data. Choose the one option which provides the best fit model (with coefficient rounded to 2 significant figures).
A) t=3.0sin(0.49d-1.6)+6.0
B) d=3.0+6.0sin(0.49t-1.6)
C) t=2.9sin(0.48d-1.5)+6.0
D) d=3.0sin(0.49t-1.6)+6.0
E) d=2.99sin(0.49t-1.60)+6.04
F) d=6.0-3.0sin(0.49t+1.6)
Can anyone help me please. I may be d?
For Further Reading
math - Count Iblis, Saturday, May 12, 2007 at 9:58am
Note that you want d(t) to be of the form:
d(t) = A + B sin(r t + s)
Now, as you can see, all the options that express d like this have r = 0.49. This makes it easy to find the best fit, because you can now transform this problem into a linear regression problem as follows. You write:
d(t) = A + B sin(r t + s) =
A + B [cos(s) sin(r t) cos(s) + sin(s)cos(rt)(rt)] =
A + B cos(s) sin(rt) + B sin(s) cos(rt)
So, this problem is of the form:
d(t) = A_1 + A_2 f_2(t) +
A_3f_3(t)
with
f_2(t) = sin(rt)
f_3(t) = cos(rt)
So this is a standard linear regression problem as d is a linear function of f_2 and f_3.
You just need to proceed in the same way as you would proceed in case of a regression problem like:
Y = A + B X1 + C X2
for given data for Y, X1, and X2.
math - bobpursley, Saturday, May 12, 2007 at 10:37am
math - Count Iblis, Saturday, May 12, 2007 at 11:51am
I've just emailed you :)
math - grant, Saturday, May 12, 2007 at 11:04am
i'm not sure what you mean. which are the correct answers?
math - Count Iblis, Saturday, May 12, 2007 at 11:49am
Hi Grant,
Do you have a calculator in which you can enter data to do linear regression in multiple variables?
If so, then you need to do the following. Compile a table with the values of X1 = sin(0.49 t) and
X2 = cos(0.49 t).
Then you must enter the data for Y = d, X1 and X2 in your calculator and out will come the regression parameters corresponding to a fit of the form:
Y = A + B X1 + C X2
From B and C you then calculate the amplitude and the phase of the sin...
math - grant, Saturday, May 12, 2007 at 12:50pm
I don't have this type of calculator as yet and my homework is due on monday. Is the answer A?
Grant, the solution is posted here:
http://www.jiskha.com/display.cgi?id=1179090779
Well, it looks like Grant had a math problem and was trying to fit a sine curve to some data. I guess he was really in deep water with this one! But don't worry, I'm here to help! After some back and forth, it seems like the answer is option E) d=2.99sin(0.49t-1.60)+6.04. So you could say that Grant finally found his answer and sailed smoothly through his homework!
To fit a sine curve to the data, we want to find an equation of the form d(t) = A + Bsin(rt + s).
From the given options, we can see that the values of r are all 0.49, so r = 0.49.
We can transform the original equation to the form d(t) = A + Bcos(s)sin(rt) + Bsin(s)cos(rt).
Now, we can treat this as a linear regression problem with f2(t) = sin(rt) and f3(t) = cos(rt).
We need to calculate the regression parameters for Y = d, X1 = f2(t), and X2 = f3(t).
Using a calculator with linear regression functionality, we can input the data for Y, X1, and X2, and calculate the regression parameters.
Unfortunately, without the data for Y, X1, and X2, we cannot provide the specific regression parameters or determine the best fit model from the given options A to F.
We suggest using a calculator with linear regression functionality to input the data and calculate the regression parameters to determine the best fit model.
In order to find the best fit model, you need to use the regression facilities on your calculator. You are looking for a sine curve of the form:
d(t) = A + B * sin(r * t + s)
By comparing the given options, you can see that all the options express d(t) in this form. The key is to determine the value of r, which is the coefficient in front of t in the sine function.
To determine the value of r, you can transform the problem into a linear regression problem. Let's rewrite the equation in a different form:
d(t) = A + B * sin(r * t + s) =
A + B * [cos(s) * sin(r * t) * cos(s) + sin(s) * cos(r * t) * (r * t)] =
A + B * cos(s) * sin(r * t) + B * sin(s) * cos(r * t)
This shows that d(t) can be expressed as a linear combination of sin(r * t) and cos(r * t) terms. Let's call sin(r * t) f2(t) and cos(r * t) f3(t). Now we have:
d(t) = A1 + A2 * f2(t) + A3 * f3(t)
This is a standard linear regression problem, with d(t) being a linear function of f2(t) and f3(t). You can proceed in the same way as you would for a regression problem like Y = A + B * X1 + C * X2, where Y, X1, and X2 are given data.
You need to compile a table with the values of f2(t) = sin(r * t) and f3(t) = cos(r * t) for the given values of t. Then, enter the data for d(t), f2(t), and f3(t) into your calculator. The calculator will provide the regression parameters corresponding to a fit of the form Y = A + B * X1 + C * X2.
From the values of B and C, you can calculate the amplitude and phase of the sine function.
Since you mentioned that you do not have a calculator with regression capabilities, you may need to solve the problem manually. Unfortunately, the solution to the problem is not provided in the information you provided. You may want to seek further assistance or try to solve the problem using other methods.