How to Solve Temperature Problems using Newton's Law of Cooling
How can we calculate the temperature of the liquid after a certain time using Newton's Law of Cooling?
Given that a pot of stew is placed on a stove to heat. The temperature of the liquid reaches 170°F and then the pot is taken off the burner and placed on a kitchen counter where the temperature of the air in the kitchen is 76°F. If k = 0.34, what will be the temperature of the liquid after 7 hours?
Calculation using Newton's Law of Cooling:
According to Newton's Law of Cooling, the rate at which a hot body loses heat is directly proportional to the difference between the temperature of the hot body and that of its surroundings.
The formula for Newton's Law of Cooling is: T(t) = Ts + (T0 - Ts)e^(-kt)
Where:
- T(t) = temperature of the body at time t
- Ts = surrounding temperature
- T0 = initial temperature of the body
- k = constant
- t = time
Substitute the given values:
Given that Ts = 76°F, T0 = 170°F, k = 0.34, and t = 7 hours
Substitute the values into the formula:
T(t) = 76 + (170 - 76)e^(-0.34*7) = 84.7°F
Newton's Law of Cooling is a fundamental principle used to determine the rate at which an object cools when exposed to a different temperature environment. In this particular scenario, we applied this law to calculate the temperature of the liquid stew after 7 hours of being off the stove.
By substituting the given values into the formula, we found that the temperature of the liquid stew after 7 hours would be 84.7°F. This calculation highlights the application of scientific principles in predicting temperature changes over time.
Understanding and applying Newton's Law of Cooling can help us analyze various cooling or heating scenarios and make informed decisions based on temperature changes. It's essential to grasp these concepts to solve temperature-related problems accurately and efficiently.