## Cool, Cooler, Coolest

Heat is constantly flowing all around us: a roaring fire transfers heat to its surroundings, a mother cow transfers heat to her cuddling calves, and heat transfers between your skin and the air that it touches.

One way that heat flows is by conduction: heat flowing through matter by way of direct contact. When you hold a mug of hot tea or a glass of ice water, heat conducts through the wall of the cup between your hand and the liquid.

The net conductive heat flow $Q$ through an object is proportional to the difference in temperatures across the object and the inherent thermal conductivity $k$ of the object, a measure of how readily the material conducts heat:

$Q \propto k\times \Delta T.$

One place you see this relationship in action is when you wrap yourself in a blanket on a chilly day. Unless you're exercising, your body releases a more or less constant flow of heat to its surroundings.

But when you wrap yourself in a blanket, you're introducing a barrier to heat flow, corresponding to a lower $k.$ In turn, less heat escapes, raising your skin temperature until the larger $\Delta T$ across the blanket balances out the lower $k.$

Since your skin temperature has risen, thermoreceptors in your skin sense the increase in temperature and you feel nice and toasty.

There are more heat transfer methods at play in the problem below than just conduction, but broadly speaking, the principles explained above still apply.

# Today's Challenge

Andrei and Cristina would like to drink some tea, but they don't want to wait too long for it to cool down. They both make tea using half a mug of boiling water.

Andrei immediately fills the rest of his mug with water straight from the fridge. Cristina does the same, but after waiting ten minutes.

Whose tea has a lower temperature now? $\\[-0.5em]$