Differential equations. Newton's law of cooling

[babe20042004]

I'm not sure how to solve this question. I'm completely lost and I dont even know where to start.

Newton’s law of cooling states that the rate at which the temperature of a hot object decreases
is proportional to the difference between the temperature of the hot object and the constant
temperature of its environment.

a. Model this law using an appropriate differential equation connecting ϴ(t) and t where
ϴ(t)is the temperature of the hot object at time t and ϴ0
is the constant temperature of
the environment.

b. Show that the general solution of the differential equation above is given by

ϴ(t)= ϴ0 + Ce^−kt



Where C and k are constants with k > 0
A hot cup of coffee in a room had an initial temperature of 85 degrees Celsius. Five
minutes later its temperature was 55 degrees Celsius. The constant temperature of the
room is 25 degrees Celsius.


c. Using the above data , determine the values of C and k in the equation

ϴ(t)= ϴ0 + Ce^−kt




d. Calculate the temperature of the cup of coffee after a further interval of five minutes.

 

[HallsofIvy]

I am puzzled by this. You have no idea how to begin? Do you at least know what the words mean?
Do you know that "the rate at which the temperature of a hot object"changes is the derivative, \(\displaystyle \frac{d\theta}{dt}\)?
Do you know that "a is proportional to b" means one is a constant multiple of the other: a= Cb for C constant?

Newton’s law of cooling states that the rate at which the temperature of a hot object decreases
is proportional to the difference between the temperature of the hot object and the constant
temperature of its environment.

says, letting \(\displaystyle \theta(t)\) be the temperature at time t, and T be "the constant temperature of its environment",
\(\displaystyle \frac{d\theta}{dt}= c(\theta- T)\)

a. Model this law using an appropriate differential equation connecting ϴ(t) and t where
ϴ(t)is the temperature of the hot object at time t and ϴ0
is the constant temperature of
the environment.

b. Show that the general solution of the differential equation above is given by

ϴ(t)= ϴ0 + Ce^−kt



Where C and k are constants with k > 0
A hot cup of coffee in a room had an initial temperature of 85 degrees Celsius. Five
minutes later its temperature was 55 degrees Celsius. The constant temperature of the
room is 25 degrees Celsius.


c. Using the above data , determine the values of C and k in the equation

ϴ(t)= ϴ0 + Ce^−kt




d. Calculate the temperature of the cup of coffee after a further interval of five minutes.