Chapter 33: Problem 24
Newton's law of cooling is given by: \(\theta=\theta_{0} \mathrm{e}^{-k t}\), where the excess of temperature at zero time is \(\theta_{0}^{\circ} \mathrm{C}\) and at time \(t\) seconds is \(\theta^{\circ} \mathrm{C}\). Determine the rate of change of temperature after \(50 \mathrm{~s}\), given that \(\theta_{0}=15^{\circ} \mathrm{C}\) and \(k=-0.02\)
Short Answer
Step by step solution
Understand the Problem
Differentiate the Function
Substitute Known Values
Calculate at t = 50 seconds
Evaluate Expression
Conclusion on Rate of Change
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Temperature Rate of Change
This is expressed by the formula \( \theta(t) = \theta_0 e^{-kt} \). Here, \( \theta(t) \) is the temperature at time \( t \), \( \theta_0 \) is the initial temperature difference, and \( k \) is a cooling constant.
The rate of change of temperature, represented as \( \frac{d\theta}{dt} \), tells us how fast the temperature is changing at any given moment. In practical terms, it's like saying how fast your coffee is cooling down after being poured.
To find the rate of change, you need to calculate the derivative of the temperature function, which involves solving a mathematical expression that includes constants and variables.
Why is it important to know the rate of temperature change? By knowing this, scientists and engineers can make predictions and consider safety measures for various processes that involve temperature regulation, such as food cooling, material manufacturing, and even weather predictions.
Exponential Functions and Their Role
When you have a negative exponent, as in \( e^{-kt} \), the function models exponential decay. This is exactly what's happening in Newton's Law of Cooling, where the temperature difference decreases over time.
Exponential decay is common in real life. It's used to explain processes like radioactive decay, population decline, and of course, cooling of objects. It's important to understand the concept of half-life, which refers to the time it takes for a quantity to reduce to half its initial value—that concept is inherently tied to exponential functions.
In simpler terms, anytime you're looking at something that decreases rapidly initially and then slows down over time, you're likely looking at an exponential decay process. Making sense of this can help you better understand temperature changes and other natural processes that follow similar patterns.
Differentiation in Action
The derivative of a function gives you the rate at which one variable changes with respect to another. For example, if you differentiate \( \theta(t) = \theta_0 e^{-kt} \), you get \( \frac{d\theta}{dt} = -k \theta_0 e^{-kt} \). Here, it tells us how quickly the temperature decreases over time.
When solving real-world problems like our temperature cooling scenario, differentiation provides insights by breaking down complex changes into simpler, more manageable rates of change. This is true across many fields, from physics to economics.
Always remember: finding the derivative is about understanding change. Whether you're dealing with temperatures cooling down or stocks fluctuating, derivatives show the instantaneous rate of how things change, giving invaluable information for decision-making, analysis, and prediction.