Apps of Calculus (Newton's Law of Cooling) Help! (2 Viewers)

[namelyanonymous]

The first part of the question is:

"A body, initially at room temperature 20 degrees celcius, is heated....."

They gave me Newton's Law of Cooling, which is:

dT/dt = -k(T-M)

where M = the surrounding temperature, T = the temperature of the body and k is just a constant.

I integrated that to eventually get:

log_e(T-M) = k(c - t)

But from the first part of the question, we know that M = 20, and at t = 0, T = 20, so if we sub those in we get:

log_e(20 - 20) = k(c)

So now we've got log_e(0), which is undefined. What am I doing wrong? :/

Thanks in advance!

 

[HeroicPandas]

 

[braintic]

If the temperature of the body is the same as the ambient temperature then there is no cooling. So either there is a mistake in the question or you have read it wrongly.

BTW the situation here is the exact opposite of that in the question you asked about projectile motion - you do NOT need to derive the result here - you can just quote T=A+Be^(kt)

 

[namelyanonymous]

If the temperature of the body is the same as the ambient temperature then there is no cooling. So either there is a mistake in the question or you have read it wrongly.

BTW the situation here is the exact opposite of that in the question you asked about projectile motion - you do NOT need to derive the result here - you can just quote T=A+Be^(kt)

The full question is:

" A body initially at room temperature 20 degrees celcius is heated so that its temperature would rise by 5 degrees/min if no cooling took place. Cooling does occur in accordance to Newton's Law of Cooling and the maximum temperature the body could attain is 120 degrees celcius. How long wold it take to reach a temperature of 100 degrees celcius."

Newton's Law of Cooling: dT/dt = -k(T - M)

I definitely think I'm understanding the question wrong in some way :/ Any idea which part?

I know I can quote it, but I really suck at remembering things so I prefer deriving everything that isn't too annoying to derive, which is most things. I tend to derive most equations, even that projectile motion one (I didn't actually ask that question (someone else initially did), I was just curious whether you were saying no you CAN derive it, because my teacher mentioned we couldn't during class, even though I would have derived it either way).

Thanks for the reply!

 

[HeroicPandas]

Translate: English -> Maths









hope this is correct

 

[namelyanonymous]

I did this, yet you still have to use the fact that at t=0, T = 20, which would lead to the whole undefined thing above.

Also, wouldn't dT/dt = 5 - k(T-M), as it is still cooling as it heats up?

Or am I still misunderstanding the first part of the question :/ ?

Edit: I'm going to try it again without using the assumption that the room itself is at 20 degrees.

 

[namelyanonymous]

Well...*ehem*...sorry about this guys but I just did a *really* silly mistake...

I forgot to add in the +5 halfway through my integration because I was doing it without thinking properly.

I missed this even though I checked over my working out like three times, and only noticed it when I redid it completely from scratch. So it was just a really stupid move on my part.


Thanks for all the replies!