Coffee Cooling Problem

Differential Equations · Axiom Academy

EXAMPLE Coffee Cooling Problem Use Newton's Law of Cooling to find the cooling constant and predict future temperatures A cup of coffee is placed in a room with ambient temperature . Its initial temperature is , and after 5 minutes it has cooled to . Using Newton's Law of Cooling, find the cooling constant k and determine how long it takes the coffee to cool to a drinkable . Excellent work! You've successfully solved a Newton's Law of Cooling problem. Here's what we learned: Newton's Law Form: the differential equation describes exponential cooling toward the ambient temperature General Solution: , where C is determined by the initial condition Finding the Cooling Constant: use one known temperature measurement to solve for k using logarithms Making Predictions: once k is known, we can predict the temperature at any future time, or find when a target temperature is reached Physical Interpretation: the negative value of k ( ) indicates cooling — a larger |k| means faster temperature change This same approach applies to many exponential decay/growth problems: radioactive decay, population growth, compound interest, and more. The key is identifying the differential equation and using initial/boundary conditions to find the constants.

This is the written version of the interactive lesson above. See the full Differential Equations course.