Problem 34 - Calculus Early Transcendentals

Question

Explain why there is exactly one constant solution to the Newton's Law of Cooling model.

Step-by-Step Solution

Answer

The constant solution is when the object's temperature equals the ambient temperature, i.e., \( T = T_{env} \).

Step 1: Understand the Newton's Law of Cooling

Newton's Law of Cooling describes how the temperature of an object changes over time as it approaches the temperature of its surrounding environment. The differential equation for Newton's Law of Cooling is given by \( \frac{dT}{dt} = -k(T - T_{env}) \), where \( T \) is the temperature of the object, \( T_{env} \) is the constant ambient temperature, and \( k \) is a positive constant rate.

Step 2: Identify the Requirement for a Constant Solution

A constant solution to a differential equation is one where the derivative of the function is zero. For Newton's Law of Cooling, this means the temperature \( T \) must not change with time, i.e., \( \frac{dT}{dt} = 0 \).

Step 3: Set Derivative to Zero

To find the constant solution, set the differential equation to zero: \( 0 = -k(T - T_{env}) \). Since \( k \) is a positive constant and not zero, this simplifies to \( T - T_{env} = 0 \).

Step 4: Solve for the Constant Solution

From the equation \( T - T_{env} = 0 \), it follows that \( T = T_{env} \). This implies the temperature of the object is equal to the ambient temperature. Since this is a condition for \( \frac{dT}{dt} = 0 \), it satisfies being a constant solution.

Step 5: Conclusion

In conclusion, the only constant solution where an object doesn't change its temperature over time is when its temperature equals the ambient temperature. Therefore, there is exactly one constant solution to the Newton's Law of Cooling, which is when \( T = T_{env} \).

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve a function and its derivatives. They are essential tools used to describe various phenomena, including the rate of change of quantities.

In the context of Newton's Law of Cooling, the differential equation is \( \frac{dT}{dt} = -k(T - T_{env}) \). This equation models how the temperature \( T \) of an object changes over time based on the temperature difference \( T - T_{env} \), where \( T_{env} \) is the ambient temperature and \( k \) is a constant.

• The left side, \( \frac{dT}{dt} \), represents the rate of change of temperature over time.
• The right side, \( -k(T - T_{env}) \), indicates that the rate of change is proportional to the difference between the object's temperature and the ambient temperature, emphasizing a restoring effect towards equilibrium.

Understanding differential equations in this way helps predict how the temperature will change until it stabilizes to a constant value.

Constant Solution

A constant solution in the realm of differential equations refers to a solution where the function's value remains unchanged over time. It is when the derivative of the function equals zero, indicating no change. For Newton's Law of Cooling, this means finding \( T \) such that \( \frac{dT}{dt} = 0 \).

Setting \( \frac{dT}{dt} = -k(T - T_{env}) \) equal to zero leads to the condition \( T - T_{env} = 0 \). Solving this, we find \( T = T_{env} \).

• This implies the object's temperature equals the ambient temperature.
• It represents a state where the temperature no longer changes, thus being constant over time.

Finding a constant solution is significant because it indicates equilibrium, where the object's temperature has stabilized with its surroundings.

Ambient Temperature

Ambient temperature, represented as \( T_{env} \) in the differential equation, is the constant temperature of the surrounding environment. It's the baseline that the temperature of an object will eventually approach as it exchanges heat.

• In Newton's Law of Cooling, the ambient temperature acts as a pivotal reference point towards which the object's temperature is drawn.
• When an object's temperature \( T \) matches \( T_{env} \), the exchange of heat ceases, and the object's temperature stabilizes, achieving a constant solution.

This highlights the importance of understanding ambient temperature as a driving force in many physical processes, such as thermal regulation and temperature balance. The intimate link between an object's temperature and its ambient environment forms the core of Newton's Law of Cooling.