Determining which coffee has a greater rate of cooling

  • Context: High School 
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    Coffee Cooling Rate
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SUMMARY

This discussion focuses on modeling the cooling rates of coffee in three different cups using exponential functions. The user successfully created a scatter plot and identified the exponential model but faced challenges in determining which coffee cools the fastest due to differing starting temperatures. The solution involves calculating the slope of each function at a common temperature point, such as 40°C, to accurately compare the rates of cooling. The user also inquires about the potential variability of the slopes at different temperatures.

PREREQUISITES
  • Understanding of exponential functions and their properties
  • Familiarity with calculus concepts, specifically derivatives
  • Ability to create and interpret scatter plots
  • Knowledge of temperature measurement in Celsius
NEXT STEPS
  • Calculate the slope of exponential functions at a common temperature point
  • Explore the concept of derivatives in relation to exponential decay
  • Investigate the effects of initial conditions on cooling rates
  • Learn about modeling real-world scenarios using mathematical functions
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Students engaged in mathematical modeling, educators teaching calculus and exponential functions, and anyone interested in the physics of thermal dynamics.

nousername
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Hey,

I have this math investigation task which asks us to model the cooling of coffee in three different cups over a period of time. Anyway, i got the data, did a scatter plot and found the exponential model, but i don't know how i could determine which one has the greatest rate of cooling.

I know that I need to look at the slope, but they don't all start from the same point on the graph, so i can't determing which one has the fastest rate of decay. If they all started from the same point i would be able to see the different functions branching off from that point, and if one has a steeper slope than the other i would know, but this wasnt the case. Can someone please help! Thanks.
 
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Calculate the slope at a common point. For example, take the point 40° at every coffee and calculate the slope at that point.
 
Hey thanks man. Do you know if the relationship of the derivative of the three graphs will change. So if i measure the derivative of each graph at some temperature (remember it's exponential) and find that one graph has a steeper slope than the other... could that change? so the other graph will now have the steeper slope? Because if that is true, then we can't use that to determine which graph has the fastest rate of cooling overall. Thanks.
 

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