Hey Guys, I have got a question provided by my teacher that I am REALLY struggling with. Could anyone please help me out?! I couldn't add an image to this post so ill try and type it out. Hopefully its readable. The question is as follows:

Newtons Law of cooling states that the rate of cooling of a body is proportional to the excess of its temperature above that of its surroundings, i.e
dT/dt = -k(T - Ts)
Where T°C is the temperature of the body, Ts °C is the temperature of the surroundings and k is %
a positive constant. A cup of hot water is poured from a kettle with an initial temperature of 95°C and is placed on a sink in the kitchen where the temperature is a constant 25°C. If it takes 10 minutes for the cup of hot water to reach a temperature of 60°C, express T in terms of t. (3 marks)

Could anyone please help me out?

7 days later

Hi, i can help with this.
first thing is to break down the information and work one at a time. so first we're given the following "initial temperature of 95°C and is placed on a sink in the kitchen where the temperature is a constant 25°C" i.e at t=0, T=95 with Ts=25.
subbing it into the formula gives:
dT/dt = -k(T-25)
Next is to flip the derivative to make t the subject when integrating.
dt/dT = -1/k(T-25)
integrating both sides gives,
t=-1/k * ln(T-25)+c where T>25.
now we can sub t=0 and T=95, which is,
0=-1/k * ln(70)+c
i.e. c=1/k * ln(70)
subbing it into the original equation gives,
t=-1/k * ln(T-25)+1/k * ln(70)
t= 1/k * ln(70/(T-25))
now we can solve for k where the new information tells us "it takes 10 minutes to reach a temperature of 60°C" which is when t=10, then T=60. subbing these in gives,
10=1/k * ln(70/(60-25))
10=1/k * ln(2)
k= ln(2)/10
finally we can sub this into our final equation of t= 1/k * ln(70/(T-25)),
t=10/ln(2)ln(70/(T-25))
but the questions asks to express T in terms of t. so just rearrange it.
(t/10)ln(2)=ln(70/(T-25))
ln(2t/10)=ln(70/T-25))
2t/10=70/(T-25)
T=70/2t/10+25 or T=70*2-t/10+25.
this can be checked where t=10 which gives T=60.

A good way to remember or understand is if u have the Cambridge textbook, go to chapter 11D. Applications of differential equations. example 15.
you can message me on discord if u need anymore help 🙂 (user is jomjom7461)

    Jom

    WOW, thank you sooo much.
    you make that sound so easy to understand.
    I really appreciate that, thanks so much for your help.

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