# Math Help - Value of k (Newton's Law of Cooling)

1. ## Value of k (Newton's Law of Cooling)

Hi
I've been given the following question;
Jane makes coffee with water from an urn which is at a temperature of 95 deg C.
Jane carries a thermometer with her at all times and notes that 7 minutes later her coffee has cooled to 70 deg C.
The room is not air-conditioned, and is 25 deg C.
Suppose the temperature follows Newton’s Law of Cooling:

where is the number of minutes after the coffee was made.
It then asks me to find A, B and k. I know my answers for A and B are correct (25 and 70 respectively) but i am struggling to find k.
It does then ask me to find the temperature after 10mins and the time for the coffee to reach 60degrees which i also have correct (62degrees and 11mins respectively). I used a value of k= ln(14/9) but it tells me this is incorrect.
Am i doing something wrong?
Thanks

2. ## Re: Value of k (Newton's Law of Cooling)

I'm not sure how you got B=70 but that is not correct.
You have 3 equations to solve for the 3 unknowns
1. Jane makes coffee with water from an urn which is at a temperature of 95 deg C.
At time t=0 the temperature is 95

2. Jane carries a thermometer with her at all times and notes that 7 minutes later her coffee has cooled to 70 deg C.
At time t=7 the temperature is 70

3. As time time tends to infinity the temperature of the coffee becomes the same as the room temperature.
when $t=\infty$, $T=25$

The equation for the third one is
$25=A+Bk^{\infty}$

There are three cases for k,
k>1 this would cause the temperature to grow to infinity as time passes which is clearly wrong
k=1 this would mean that T(t)=A+B which is constant, clearly temperature is not constant so k cannot be 1
k<1 is the only reasonable case left

If k<1 then $k^{\infty}=0$ so $25=A+Bk^{\infty}$ simplifies to $25=A$

Now you have A, use the other two equations to find B and k. Once you have them the second part of your question should follow easily.