Number of hours of daylight - Periodic functions.

[Evangeline101]

Homework Statement

Homework Equations

none

The Attempt at a Solution

a) It is a periodic relationship because the number of hours of daylight repeats each year?
OR
It is a periodic relationship because the number of hours of daylight is based on the rotation of the earth, which is also periodic..?

b) The period of this function is 360 days because the function repeats every 360 days.

c) For day 280, the number of hours of daylight would be about 12 hrs:

d) For day 320, the number of hours of daylight would be about 6 hrs:

(For the graphs, should i draw a line of best fit?)
I would really appreciate it if someone could look over my answers and make sure everything is in order!
Thanks!

 

[James R]

a) It is a periodic relationship because the number of hours of daylight repeats each year?
OR
It is a periodic relationship because the number of hours of daylight is based on the rotation of the earth, which is also periodic..

How long does it take for the Earth to rotate on its axis once? What relation does that have to daylight and darkness?

You say the number of hours of daylight repeats over a yearly cycle, which makes sense. You get more daylight hours in summer than in winter. Do you know why that is the case?

b) The period of this function is 360 days because the function repeats every 360 days.

Is 360 days related to the rotation of the Earth, or something else?

For parts (c) and (d), your answers look to be on the right track, but the points you have plotted look a little bit off to me, just judging by eye.

You ask about drawing a line of best fit. Are you familiar with sine and cosine curves?

 

[SammyS]

Homework Statement

View attachment 103088
View attachment 103089
View attachment 103090

Homework Equations

none

The Attempt at a Solution

a) It is a periodic relationship because the number of hours of daylight repeats each year?
OR
It is a periodic relationship because the number of hours of daylight is based on the rotation of the earth, which is also periodic..?

b) The period of this function is 360 days because the function repeats every 360 days.

c) For day 280, the number of hours of daylight would be about 12 hrs:
View attachment 103092

d) For day 320, the number of hours of daylight would be about 6 hrs:
View attachment 103093
(For the graphs, should i draw a line of best fit?)

I would really appreciate it if someone could look over my answers and make sure everything is in order!
Thanks!

There was a thread started late in May of this year with the same problem to solve.

Periodic functions​

OP never responded to the few replies there, but there may be a small amount of help for you there.

Both your answers to part a) are correct, with the second reason being the basis for the first of your answers.

You may find it interesting to "Google" Yellowknife. You'll find that it's a city in Canada.Your answer(s) to part a) should give you the answer to part b), without resorting to the points plotted on the graph.

How many days are there in a year in Canada?

 

[Evangeline101]

You ask about drawing a line of best fit. Are you familiar with sine and cosine curves?

Yes I am familiar with these curves, I am pretty sure that this graph represents a cosine curve.

For parts (c) and (d), your answers look to be on the right track, but the points you have plotted look a little bit off to me, just judging by eye.

Could you suggest a way to make the points more accurate?

 

[Evangeline101]

Both your answers to part a) are correct, with the second reason being the basis for the first of your answers.

Okay, so I could say:
It is a periodic relationship because the number of hours of daylight repeats each year, also the number of hours of daylight is based on the rotation of the Earth which is also periodic.

You may find it interesting to "Google" Yellowknife. You'll find that it's a city in Canada.

Yes, I should know this since I am Canadian

Your answer(s) to part a) should give you the answer to part b), without resorting to the points plotted on the graph.

How many days are there in a year in Canada?

Okay so the period would be 365 days since the number of hours of daylight repeats each year...?

 

[SammyS]

...

Okay so the period would be 365 days since the number of hours of daylight repeats each year...?

Yes.

 

[James R]

Yes I am familiar with these curves, I am pretty sure that this graph represents a cosine curve.

Sine and cosine curves have the same shape. In a sense they are really the same function, just shifted along the axis relative to one another.

Could you suggest a way to make the points more accurate?

As the link to the previous thread on this question shows, the easy way is to use the symmetry that you expect from the curve over the course of the year. For example, the largest number of daylight hours will be at the summer solstice. One day before and after that the number of daylight hours will be equal to one another. 10 days before and after will be equal. 100 days before and after will be equal, and so on.

A more "algebraic" way of completing the plot would be to work out the parameters (Amplitude, phase constant, period) of the particular sinusoidal function on the graph and then use that to calculate the expected value at any given day. However, probably the question does not expect you to do that. Also, before you could actually justify doing it that way you'd really need to know why the curve should be a sine/cosine shape in the first place, which is probably beyond what the question is expecting you to know.

 

[James R]

Okay, so I could say:
It is a periodic relationship because the number of hours of daylight repeats each year, also the number of hours of daylight is based on the rotation of the Earth which is also periodic.

Day and night are caused by the Earth's rotation. The Earth rotates once every 24 hours, so it gets approximately 12 hours of darkness and 12 hours of daylight each day. However, it's not exactly 12 hours of each because the Earth's axis is tilted relative to the plane of the Earth's orbit around the sun. It is that tilt that causes the seasons and also causes the number of hours of daylight and darkness to change over the course of a year. In Canada, when the North pole of the Earth is pointing roughly in the direction of the Sun, it is Summer and you get more daylight because the northern hemisphere spends more of each 24 hour day in the sunlight. On the other hand, when the North pole is tilted directly away from the Sun, it is Winter and you get less daylight because the northern hemisphere spends more time in the "shadow" away from the Sun.

The actual number of daylight hours varies with latitude. The maximum variation over the course of year happens at the poles of the Earth, and the least variation happens at the equator. For example, the North pole and South pole each have total darkness for several months of the year (the sun is never seen to rise above the horizon during those months), and constant daylight for another several months (when the sun never sets).

 

[Evangeline101]

Extremely sorry for the late reply ( I was studying for an exam and didn't have time to focus on this) but now I am back :)

For parts (c) and (d), your answers look to be on the right track, but the points you have plotted look a little bit off to me, just judging by eye.

Ok so I attempted part c and d again, and wanted to know if these points are plotted more accurately:

c)

d)

On the same graph, it looks like this:

 

[SammyS]

Extremely sorry for the late reply ( I was studying for an exam and didn't have time to focus on this) but now I am back :)

Ok so I attempted part c and d again, and wanted to know if these points are plotted more accurately:
c)
View attachment 104060

d)
View attachment 104061

On the same graph, it looks like this:
View attachment 104062

They look reasonable. Can you explain how you arrived at those results?

 

[Evangeline101]

They look reasonable. Can you explain how you arrived at those results?

Well, i am not sure if this is a proper way to do it, but for part c) i took the 80 day point and used the same point for 280 days (which is about 11.8), and for part d) i took the 40 day point and used the same point for 320 days (which is about 7.2), I don't know if that makes any sense?

 

[SammyS]

Well, i am not sure if this is a proper way to do it, but for part c) i took the 80 day point and used the same point for 280 days (which is about 11.8), and for part d) i took the 40 day point and used the same point for 320 days (which is about 7.2), I don't know if that makes any sense?

After a second look, and your explanation, no those points don't look quite right.

Half a year is 182.5 days.

The summer solstice is on or about June 21. That's day 172. That agrees with the graph. The graph should be symmetric with respect to day 172.

 

[Evangeline101]

Ok so from my understanding:

for part c) the number of hours of daylight on day 280 would be about 10.1 hrs:
(the green line roughly shows the line of symmetry at 172 days)

part d) the number of hours of daylight on day 320 would be 6 hrs:

On the same graph:

Is this an improvement?

 

[SammyS]

In my opinion, this is the improvement you needed.

Looks good.

 

[Evangeline101]

In my opinion, this is the improvement you needed.

Looks good.

Ok, is it better to draw a curve through the points or just leave it as points?

 

[Evangeline101]

Also, for part b) would the period of the function possibly be 340 days, if you continue to extend the pattern in this way it stops at 340 days and then the pattern would repeat from this point...?

 

[SammyS]

Also, for part b) would the period of the function possibly be 340 days, if you continue to extend the pattern in this way it stops at 340 days and then the pattern would repeat from this point...?

No.

You can't determine this from the graph.

What is the basic cause for the number of hours os daylight being periodic?

 

[Evangeline101]

What is the basic cause for the number of hours os daylight being periodic?

It repeats each year, so it would stay at 365 days.

 

[SammyS]

It repeats each year, so it would stay at 365 days.

Correct !

 

[Evangeline101]

Correct !

Ok thank you so much for the help, I really appreciate it!