• Show how quantities change using derivatives and explore how these changes are connected
• Apply the chain rule to calculate how one changing quantity affects another

Coffee Cooling Rate Analysis: Exploring Related Rates

In this apply-it task, we’ll investigate the cooling rate of coffee in various containers, applying concepts of derivatives and the chain rule to real-world scenarios. This will help us understand how different factors affect the rate of temperature change over time.

 

Given: The temperature of coffee $T(t)$ over time $t$ is given by: $T(t) = T_{a} + (T_{0} - T_{a})e^(-kt)$, where:

• $T(t)$ is the temperature of coffee at time t min,
• $T_{a}$ is the ambient temperature,
• $T_{0}$ is the initial temperature of the coffee,
• $k$ is the cooling constant specific to the container,
• $t$ is time in minutes.