# Math Help - Newton's law of cooling/warming

1. ## Newton's law of cooling/warming

A thermometer reading 4˚C is brought into a room where the temperature reading is 30˚C. If the thermometer reads 10˚C after 2 minutes, determine the temperature reading 5 minutes after the thermometer is first brought into the room.

2. 22.46°c?

3. Originally Posted by mikegar813
22.46°c?

4. 30+(4-30)e^(-k2)
k=-.5ln (10/13)
30+(-26)e^((-3)-.5ln (10/13))=12.46
2 minutes = 10°C
3 minutes = 12.46°C
10+12.46= 22.46°C

is this correct?

5. Originally Posted by mikegar813
30+(4-30)e^(-k2)
k=-.5ln (10/13)
30+(-26)e^((-3)-.5ln (10/13))=12.46
2 minutes = 10°C
3 minutes = 12.46°C
10+12.46= 22.46°C

is this correct?
I have no idea what this all means. Where have you used Newton's law of cooling/warming to set up the differential equation? Where is your working for solving that differential equation?

6. H= temperature of the object at time t
H sub s= the constant surrounding temperature
t= time
H sub 0= the temperature at t=0
H-H sub s=(H sub 0-H sub s)e^(-kt)

I found this formula in my book.

7. So you didn't actually use "Newton' law of cooling", you used a formula derived from that.

Okay, here $H_0= 4$, $H_s= 30$ and your formula becomes $H= (4- 30)e^{kt}$. You want to find H when t= 5 but you still don't know what "k" is. You can find that by using the fact that H= 10 when t= 2: $10= (4- 30)e^{2k}$. Then you can find H when t= 5 by using that same k and t= 5.

Warning: This is NOT linear! H(5) is NOT equal to H(2)+ H(3)! Do NOT find H(3), just use the formula with t= 5, not 3.

8. so 'k' would not change even though the 't' time does?

9. Originally Posted by mikegar813
so 'k' would not change even though the 't' time does?
k is a constant. It does not change.

Were you expected to use Newton's Law of Cooling to solve this question? If yes, then the fact that k is a constant should be clear and you shouldn't be taking shortcuts by applying formulas you don't understand. If not, then the title of your post is misleading and a lot of time might have been wasted answering this question in a way that would ultimately have made no sense to you.