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Chapter 10: Problem 20

A cool object is placed in a room that is maintained at a constant temperature of \(20^{\circ} \mathrm{C}\). The rate at which the temperature of the object rises is proportional to the difference between the room temperature and the temperature of the object. Let \(y=f(t)\) be the temperature of the object at time \(t ;\) give a differential equation that describes the rate of change of \(f(t)\)

Short Answer

Expert verified
The differential equation is \( \frac{df(t)}{dt} = k (20 - f(t)) \).

Step by step solution

01

Identify the variables

Let the temperature of the object at time t be denoted by the function, \( y = f(t) \). The room temperature is a constant \( 20^{\bullet} \text{C} \).
02

Understand the given information

The problem states that the rate of change of the object's temperature is proportional to the difference between the room temperature and the object's temperature.
03

Express the difference in temperatures

The difference between the room temperature and the object's temperature is \( 20 - f(t) \).
04

Formulate the proportional relationship

Since the rate of change of the object's temperature (\( f(t) \)) is proportional to this difference, we can write: \[ \frac{df(t)}{dt} = k (20 - f(t)) \] where \( k \) is the proportionality constant.
05

Write the differential equation

The differential equation that describes the rate of change of \( f(t) \) is: \[ \frac{df(t)}{dt} = k (20 - f(t)) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical equations that relate a function with its derivatives. In simpler terms, they describe how a particular quantity changes over time. They are widely used in various fields such as physics, engineering, and biology to model real-world scenarios. In the context of the Newton's Law of Cooling problem, the differential equation describes how the temperature of an object changes over time.
Here, the function representing the temperature of the object is denoted by \( y = f(t) \). The differential equation we derived is: \( \frac{df(t)}{dt} = k (20 - f(t)) \). This tells us how fast the object's temperature changes at any moment in time.
Proportional Relationship
A proportional relationship means that one quantity changes at a constant rate with respect to another. In Newton's Law of Cooling, the rate of change of the object's temperature is directly proportional to the difference between the room temperature and the object's current temperature.
If we translate this into mathematical terms, we get the expression: \( \frac{df(t)}{dt} = k (20 - f(t)) \), where:
  • \( \frac{df(t)}{dt} \) represents the rate of change of the object's temperature
  • \( k \) is the proportionality constant
  • \( 20 - f(t) \) is the difference between the room temperature and the object's temperature
This equation encapsulates the idea that the bigger the difference between the room temperature and the object's temperature, the faster the temperature of the object will change.
Temperature Change
Temperature change in this context refers to how the temperature of the object evolves over time as it comes closer to the room temperature. According to Newton's Law of Cooling, the object will gradually reach thermal equilibrium with the surrounding environment.
To understand this better, let鈥檚 revisit our differential equation: \( \frac{df(t)}{dt} = k (20 - f(t)) \). This equation shows that:
  • When the object's temperature \( f(t) \) is much lower than the room temperature (20掳C), the rate of temperature change \( \frac{df(t)}{dt} \) will be higher, meaning the object heats up quickly.
  • As the object's temperature \( f(t) \) approaches 20掳C, the rate of change \( \frac{df(t)}{dt} \) decreases, meaning the object heats up more slowly.
This process continues until the temperature of the object matches the room temperature, at which point the rate of change becomes zero, signifying no further temperature change.

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Most popular questions from this chapter

Solve the following differential equations with the given initial conditions. $$ \frac{d y}{d x}=\frac{\ln x}{\sqrt{x y}}, y(1)=4 $$

Let \(f(t)\) be the solution of \(y^{\prime}=-(t+1) y^{2}, y(0)=1\). Use Euler's method with \(n=5\) to estimate \(f(1) .\) Then, solve the differential equation, find an explicit formula for \(f(t)\) and compute \(f(1)\). How accurate is the estimated value of \(f(1)\) ?

One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a \(y z\) -graph if one is not already provided. Always indicate the constant solutions on the \(t y\) -graph whether they are mentioned or not. \(y^{\prime}=5 y-y^{2}, y(0)=1, y(0)=7\)

Suppose that \(f(t)\) is a solution of the differential equation \(y^{\prime}=t y-5\) and the graph of \(f(t)\) passes through the point \((2,4)\). What is the slope of the graph at this point?

The Federal Housing Finance Board reported that the national average price of a new one-family house in 2012 was $$\$ 278,900$$. At the same time, the average interest rate on a conventional 30 -year fixedrate mortgage was \(3.1 \% .\) A person purchased a home at the average price, paid a down payment equal to \(10 \%\) of the purchase price, and financed the remaining balance with a 30 -year fixed-rate mortgage. Assume that the person makes payments continuously at a constant annual rate \(A\) and that interest is compounded continuously at the rate of \(3.1 \% .\) (Source: The Federal Housing Finance Board, www.fhfb.gov.) (a) Set up a differential equation that is satisfied by the amount \(f(t)\) of money owed on the mortgage at time \(t\) (b) Determine \(A\), the rate of annual payments, that is required to pay off the loan in 30 years. What will the monthly payments be? (c) Determine the total interest paid during the 30 -year term mortgage.
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