Car Talk Puzzler: Old Fashioned Coffee

I recently came across a Car Talk puzzler that has me befuddled. The answer is also confusing, and at first glance, it may seem incorrect. So, I want to try to dive in-depth into answering the puzzler on my own and have some fun with it.

For the New Kids

Recently, I started listening to the now-ended NPR radio show called "Car Talk." To describe it to someone who has never heard of the show before: it's a show where two brothers, Tom and Ray Magliozzi, who are mechanics, answer car and car repair questions from callers. However, it stands out due to the humor and banter the hosts bring to the show through mail-in letters, skits, and puzzler segments. Many of the call-ins are not even about specific car repair questions, instead centering around interpersonal situations and minutia of everyday life. Self-described as "... it's a little bit about cars, and the rest is who knows".

Many people my age, the 20-25-year-old demographic, either have never heard of the show or vaguely remember it if their parents used to listen to "Car Talk" as well. Indeed, most people under this age range or from outside the US may have never heard of the show at all, which is something I hope to share with them. If you haven't heard the show before, listen to it.

I remember listening to "Car Talk" from when I was younger, and my dad would drive me to the Florida Atlantic University math circles program on Saturdays. Now, I am at a different stage in my life, working on my Ph.D. and living my own young adult life. In many ways, my interest still lies in engineering, science, and math.

The Puzzler

The core idea of the puzller deals with cooling a cup of coffee. I have paraphased it from the car talk puzzler website as accuracly as I can below.

You just finsehd boiling a cup of coffe and pour the freshly brewed and hot coffe into your cocffee mug. You also like to take your coffe with creamer.

Since the coffee is hot you want to let it cool for 5 minutes. However, since you also want creamer you now have two options.

• Option 1: You can pour the creamer into the coffee right away and let the coffee cool for 5 minutes.
• Option 2: You can wait 5 minutes and then pour the creamer into the coffee.

Which apprach makes the coffee the coolest?

The claimed answer is shown below. This is direcly quoted from the car talk puzzler website.

It is better to wait the 5 minutes before adding the cream.

So in order to get the fastest cooling for the coffee, you want the greatest temperature difference between the hot stuff and the cold stuff.

So if you add the cream first, you lower the temperature of the coffee and this reduces the rate in which it cools down. So adding the cream first might lower the temperature somewhat at first, what you actually do is slow the cooling process completely. The temperature difference won't be the greatest if the cream is added early.

It is called delta T. Heat transfer equation.

A real puzzler that time!

At first glkanc the answer doesnt make sense to me at all and I feel like I dont get the underlaying inution. It might be harder since the language is a bit vauge and lacks some context. This is my motivation to uncover what is really going on.

The Source

One major issue is the fact that the above question and answer were adapted form the car talk puzzler website which itslef is a pharapshing of the puzller as it weas read on air. This is going to cause a point of ambiguity.

I remeber listening to the puzller and I belive they mention that the creamer is taken out and set on the counter at the beginnig of the 5 minuets. This might be mentiedn off hand as not a signifiant detal part of the orginal concoaial puzller desciption. However, we will hold on to this modifcation to the puzzler for later analaysis. For now I will do an analysis where the creamer is not taken out of the fridge untill it is time to add it to the coffee, either at the begining or end of the 5 minuets.

High School Science

Thking back to whne I took physics and chemtny in high school, it seems like we can model the problem and possibly even solve it on paper. Easy!^[1]

The starting point is "Newton's law of cooling".

The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment.

This can be modeled differntial equation shown below:

$$\frac{dT}{dt}=-k(T-T_{env})$$

Solving for a closed form solution we get:

$$T(t)=T_{env}+(T_0-T_{env})e^{-kt}$$

Where $T_0$ is the initial temperature of body and $T_{env}$ is the temperature of the environment. This models an exponential decay of the temperature of the body towards the temperature of the environment. The most imporant factor here is the constant $k$ which is the rate of decay. This varys due to the physcal properis of the mateial and how heat is transfered in a physcal envirmnmt. We will come back to this later.

Newton's law of colling will allows us to model how the cofee will decay over time.

Now we need one more tool to model how the mnixing of coffee and creamer will affect the temperature of the combined mixture. Lets model this using thermal energy and the conservation of energy.

The chnage in thermel energy of an substance can be defined in one way shown below:

$$Q=mc\mathrm{\Delta }T$$

Where $Q$ is the chnage in thermel energy, $m$ is the mass, $c$ is the specific heat, and $\mathrm{\Delta }T$ is the change in temperature.

$Q$ is also known as the heat and is the amount of energy needed to raise the temperature of the substance by the tempture $\mathrm{\Delta }T$.

This quation also makes snese in the context of abilaute measurem,ent of energy as oopsed to tempature alone since it also incoraptes mass. For example, if I mix a large bathtub of hot water with a small cup of cold water final mizuture will remain pretty hot. However if I use more cold water the final mixture will be cooler. This intution makes sense to me and it seems like it could help understadn what the answer to the puzzler could be. However, since we add the same amount of coffee and creamer to scenairo A and B, its makes it confusing to see how mass will play a role in the answer in the context of the cofffee senaoir.

We can then apply the conservation of energy to the coffee and creamer mixture as shown below:

$$\begin{array}{c}{E}_{\text{coffee}}+E_{\text{coffee}}=E_{\text{mixture}}\\ Q_{\text{coffee}}+Q_{\text{coffee}}=Q_{\text{mixture}}\\ m_{\text{coffee}}{c}_{\text{coffee}}\mathrm{\Delta }{T}_{\text{coffee}}+m_{\text{coffee}}{c}_{\text{coffee}}{T}_{\text{coffee}}=(m_{\text{coffee}}+m_{\text{coffee}})c_{\text{mixture}}{T}_{\text{mixture}}\end{array}$$