# Coffee and milk problem

## Homework Statement

The coffee in a cup is at a temperature T(to) when t=to in a room that has temperature T1=20 degrees Celsius. The temperature of the coffee is found using the function:
$$T(t) = (T(t_0) - T_1) e^{(-\frac{t-t_0}{10})} + T_1 , t >= t_0$$

We add milk, so that the cup contains 90% aforementioned coffee and 10% milk. We are given that T(to) = 100 degrees and that we plan to drink the coffee at t=10. Refrigerated milk has 5 degree temperature. What's the best moment to add the milk so that when we decide to drink our coffee it has the highest possible temperature??

## Homework Equations

$$T(t) = (T(t_0) - T_1) e^{(-\frac{t-t_0}{10})} + T_1 , t >= t_0$$

## The Attempt at a Solution

There are reasons why I'm having trouble figuring my way through the problem, and mostly it's because I haven't ever dealt with temperature of a mixture of liquids. I know that I must construct a function of time that should produce max at t=10 but that is not all. Is it correct to consider that temperature of the cup will be 0.9 T(t) and 0.1 * 5, like a friend proposed? I would very much appreciate a hint, thanks!

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Milk and coffee are mostly water so you can assume identical specific heats here. To compute the temperature of the mixture, mass average the temperatures.

T = (Mcoffee(T) + Mmilk(T))/(Mcoffee + Mmilk)
where Mmilk = .1(Mcoffee)

This will yield a slightly different result from what your friend suggested.

Milk and coffee are mostly water so you can assume identical specific heats here. To compute the temperature of the mixture, mass average the temperatures.

T = (Mcoffee(T) + Mmilk(T))/(Mcoffee + Mmilk)
where Mmilk = .1(Mcoffee)

This will yield a slightly different result from what your friend suggested.
Karkas said:
We add milk, so that the cup contains 90% aforementioned coffee and 10% milk
actually I corrected the description, my mistake.

Andrew Mason
Homework Helper

## Homework Statement

The coffee in a cup is at a temperature T(to) when t=to in a room that has temperature T1=20 degrees Celsius. The temperature of the coffee is found using the function:
$$T(t) = (T(t_0) - T_1) e^{(-\frac{t-t_0}{10})} + T_1 , t >= t_0$$

We add milk, so that the cup contains aforementioned coffee and 10% milk. We are given that T(to) = 100 degrees and that we plan to drink the coffee at t=10. Refrigerated milk has 5 degree temperature. What's the best moment to add the milk so that when we decide to drink our coffee it has the highest possible temperature??

## Homework Equations

$$T(t) = (T(t_0) - T_1) e^{(-\frac{t-t_0}{10})} + T_1 , t >= t_0$$

## The Attempt at a Solution

There are reasons why I'm having trouble figuring my way through the problem, and mostly it's because I haven't ever dealt with temperature of a mixture of liquids. I know that I must construct a function of time that should produce max at t=10 but that is not all. Is it correct to consider that temperature of the cup will be 0.9 T(t) and 0.1 * 5, like a friend proposed? I would very much appreciate a hint, thanks!
The specific heat of coffee or milk should be the same as the specific heat of water and is temperature-independent in this temperature range.

To minimize the temperature drop over the time interval, you have to minimize the rate at which temperature changes. (this is proportional to the rate at which heat flows to the room).

So work out the change in temperature as a function of time and see how it depends on the initial temperature. It appears that $\dot T \propto (T(t_0) - T_1)$. So how would you minimize the rate of temperature change (ie the rate at which heat flows to the room)?

AM

Thanks for the replies:

@Andrew Mason: You're suggesting I should minimize the rate of temperature change, that is T'(t), so I should consider T''(t) = 0?

Isn't it true that from t0 to t=10 the temperature of the content of the cup will go through 2 phases? From t0 till the point that we add the milk it will be T(t) and from that point till t=10 it will be the weighted average of T(t) and Tmilk.

How can we incorporate the time point that we add milk to the equation of temperature of the mixture? Will we change the time variable of T to t' , with t' being the point that we add the milk and then proceed to differentiate?

I hope my thoughts don't sound too scrambled...

Andrew Mason