The busy Mother pours herself a coffee into a paper cup before making her way to the amusement park. The coffee temperature is 35 C when the cup is placed on the kitchen counter. The Mother needs to tend to her son and her coffee is forgotten. When the Mother finally returns to her coffee 35 minutes later, the temperature is now 21 C.

Question

a.what type of function best models the cooling of a hot liquid?
b. What is the mathematical model for this situation? (i.e. - the equation)
c. If the optimal temperature for drinking a hot liquid is 280C, at what time would the mother have had to return in order to enjoy her cup of coffee?

Answer 1

recall Newton's Law of Cooling states that

T(t) = T₀ - (T₀-T_a)e^(-kt)
Unfortunately, you have not specified T_a (the ambient temperature), so all we have so far is

T(t) =35 - (35-T_a)e^(-kt)
and you know that T(35)=21

You can see that the cooler the room is, the faster the coffee will cool.

On the other hand, maybe you are using a linear cooling model, in which case

T(t) = 35 - (35-21)/35 t = 35 - 2/5 t
If so, then you want t such that T(t) = 28

Answer 2

I need to solve this question using math (advanced functions), not physics.