• 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 [latex]T(t)[/latex] over time [latex]t[/latex] is given by: [latex]T(t) = T_{a} + (T_{0} - T_{a})e^(-kt)[/latex], where:

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