Problem 43
Question
Surrounding medium of unknown temperature A pan of warm water \(\left(46^{\circ} \mathrm{C}\right)\) was put in a refrigerator. Ten minutes later, the water's temperature was \(39^{\circ} \mathrm{C} ; 10\) min after that, it was \(33^{\circ} \mathrm{C}\) . Use Newton's Law of Cooling to estimate how cold the refrigerator was.
Step-by-Step Solution
Verified Answer
Refrigerator temperature is estimated at 26°C.
1Step 1: Understand Newton's Law of Cooling
Newton's Law of Cooling describes the rate of cooling of an object. It states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature.
2Step 2: Set Up the Differential Equation
According to Newton's Law of Cooling, the equation is \( \frac{dT}{dt} = -k (T - T_a) \) where \( T \) is the temperature of the object at time \( t \), \( T_a \) is the ambient temperature, and \( k \) is a constant. We are given \( T(t=0) = 46^{\circ}C \), \( T(t=10) = 39^{\circ}C \), and \( T(t=20) = 33^{\circ}C \).
Key Concepts
Differential EquationsAmbient TemperatureRate of CoolingTemperature Change
Differential Equations
Differential equations are mathematical tools used to analyze how things change over time or space. In the context of Newton's Law of Cooling, a differential equation describes how the temperature of an object changes as it approaches the ambient temperature. The equation used is: \[\frac{dT}{dt} = -k (T - T_a)\]Here, \(T\) represents the temperature of the object, \(T_a\) is the ambient temperature, and \(k\) is a constant that indicates the rate of cooling. This formula helps to predict how quickly an object will adjust to the temperature of its surroundings.
- \(\frac{dT}{dt}\) means the rate of temperature change over time.
- \(-k (T - T_a)\) indicates the proportional difference between the object's temperature and ambient temperature.
Ambient Temperature
Ambient temperature refers to the temperature of the surrounding environment. In the context of cooling, it is the temperature to which an object aims to stabilize over time. Using Newton's Law of Cooling, the ambient temperature influences how quickly an object like a pan of warm water will cool down.
In our scenario, we don't know the refrigerator's temperature, which is the ambient temperature we wish to determine. As the water cools, it approaches this unknown temperature. Since the ambient temperature is a constant in our equations, determining its value is crucial for predicting the cooling process effectively.
In our scenario, we don't know the refrigerator's temperature, which is the ambient temperature we wish to determine. As the water cools, it approaches this unknown temperature. Since the ambient temperature is a constant in our equations, determining its value is crucial for predicting the cooling process effectively.
Rate of Cooling
The rate of cooling describes how quickly something cools down, or more technically, the speed at which its temperature decreases. Newton's Law of Cooling expresses this rate through a constant \(k\), which shows how responsive an object is to environmental temperature changes.
The value of \(k\) depends on various factors like the material of the object, the heat transfer conditions, and the difference between the object's temperature and the ambient temperature. In our example, a larger \(k\) would mean the water reaches the refrigerator's temperature more quickly. Solving the differential equation helps us find this rate and better understand the cooling dynamics involved.
The value of \(k\) depends on various factors like the material of the object, the heat transfer conditions, and the difference between the object's temperature and the ambient temperature. In our example, a larger \(k\) would mean the water reaches the refrigerator's temperature more quickly. Solving the differential equation helps us find this rate and better understand the cooling dynamics involved.
Temperature Change
Temperature change in the context of Newton's Law of Cooling is the transition of an object's temperature from an initial state to equilibrium with the ambient environment. In our problem, the initial temperature is \(46^{\circ} C\), and the goal is to find how it changes over time to reach the ambient temperature of the refrigerator.
We observe the water's temperature at steady time intervals:\(39^{\circ} C\) after 10 minutes, dropping to \(33^{\circ} C\) after another 10 minutes. These measurements help us apply Newton's Law of Cooling effectively, allowing us to plot the temperature change over time and estimate unknowns such as the refrigerator's ambient temperature. Understanding this process enables us to predict similar cooling patterns in different scenarios.
We observe the water's temperature at steady time intervals:\(39^{\circ} C\) after 10 minutes, dropping to \(33^{\circ} C\) after another 10 minutes. These measurements help us apply Newton's Law of Cooling effectively, allowing us to plot the temperature change over time and estimate unknowns such as the refrigerator's ambient temperature. Understanding this process enables us to predict similar cooling patterns in different scenarios.
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