A REDICULOUSLY hard questions about building a temperature model

[qwerty11]

Please help. I need this answered by tomorrow morning.

The question is:
Compute the average temperature over the Earth's surface given the fact that the Earth is a sphere with radius a=p. The temperature model is given by the linear transformation T=2(deg)+85sin(φ), with a 90deg rotation. Use spherical coordinates to compute the result.

Let x=asinucosv, y=asinusinv, and z=acosu. Assume the poles is 2deg and the temperature along the equator is 85deg facing the sun. The average temperature over the Earth's surface is the surface integral of T divided by the surface area.

1. Compute the surface are of a sphere x^2+y^2+z^2=a^2


(I had no problem with this using a double integral. Found it to be 4a^2π)


2. Compute the base area.


(No problem with this. Found it to equal to a^2sinu or p^2sinφ


3. Compute the surface integral.


4. Compute the average temperature of the Earth's surface using the given linear transformation.


5. Write the model that computes the Earth's average temperature given the temperatures along the poles and equator.








Thanks for the help!

 

[betel]

So where is your problem? Do you not know how to calculate the surface integral or do you have problems to state the temperature transformation correctly?

 

[qwerty11]

I am having difficulty figuring out both unfortunately.

 

[betel]

Ok. then let's see if we can figure out how the temperature looks like on Earth in this model.
If it wasn't for the part with "equation facing the sun" I would say, the temperature is given your formula with $$\phi$$ the polar angle and symmetric around the axis.

The part with the equator confuses me though. First I think it should be 87deg, but the part with facing the sun is strange.

Was there any specification for the angle $$\phi$$ given, or do you have any common use in the lecture?

 

[qwerty11]

Unfortunately no. This is all the information I was given.

 

[betel]

Well, then i would say you have to integrate T=2+85 sin u, over the sphere. Can you do this?

 

[qwerty11]

I do not believe so. I am totally lost for parts 3, 4, & 5.

 

[betel]

Well for three you have to calculate $$\int (2+85 \sin u)p^2\sin u du dv$$.
For 4 you have to divide this by the surface area of the earth.
I think for 5 you have to write the general expression without fixed values for the 2 and 85. I.e. so you can enter the values later.

 

[qwerty11]

No bound for that integral?

 

[betel]

Of course with the correct bound. It just a pain to write it every time.

 

[qwerty11]

The same ones I used for part one? 2pi over 0 and pi over 0?

 

[betel]

Yes. You still integrate on the same surface.

 

[qwerty11]

How would you integrate the p^2 since there is no variable for that integration?

 

[betel]

Then it's a constant factor.

 

[qwerty11]

Ok so I have 4pi^2-170pi for 3 correct?

 

[betel]

Check for yourself. Is this result sensible?170pi is much greater than 4 pi^2, so the total will be negative. Your temperature is positive everywhere so somewhere you must have miscalculated.

 

[qwerty11]

Im majorly confused. I have attached a sheet of my calculations. If you could just point out my error I would like to fix it.

 

[betel]

First: in your calculation you missed the cos 0.
Second: cos pi is -1 not 1
Third you picked the wrong integrals. Check my integral again. u correspond to theta and v to phi. The bound for u are 0 to pi. For v are 0 to 2pi.
Do not forget the extra sin theta.