Question
What does the horizontal asymptote represent in the graph of T(t)=S+(T subscript --S)e^kt for Newton's Law of Cooling?
Answers
Answered by
GPT-4o mini
In the context of Newton's Law of Cooling, the equation you provided likely represents the temperature \( T(t) \) of an object at time \( t \), where \( S \) is the surrounding temperature (the ambient temperature), \( T_S \) is the initial temperature of the object, \( k \) is a constant that depends on the characteristics of the cooling process, and \( e \) is the base of the natural logarithm.
The equation can be expressed as:
\[
T(t) = S + (T_S - S)e^{kt}
\]
As \( t \) approaches infinity (\( t \to \infty \)), the term \( e^{kt} \) behaves differently depending on the sign of \( k \):
- If \( k < 0 \), \( e^{kt} \) approaches 0 as \( t \) increases, meaning \( T(t) \) approaches \( S \).
- If \( k > 0 \), then \( T(t) \) approaches infinity, which isn't typically relevant in cooling scenarios.
In most cooling scenarios, we assume \( k < 0 \). Therefore, the horizontal asymptote represents the surrounding (ambient) temperature \( S \), indicating that as time goes on, the temperature of the object \( T(t) \) will asymptotically approach the surrounding temperature. Essentially, this means that over a long period of time, the object's temperature stabilizes at the ambient temperature, reflecting the idea that the object cools down to the temperature of its surroundings.
The equation can be expressed as:
\[
T(t) = S + (T_S - S)e^{kt}
\]
As \( t \) approaches infinity (\( t \to \infty \)), the term \( e^{kt} \) behaves differently depending on the sign of \( k \):
- If \( k < 0 \), \( e^{kt} \) approaches 0 as \( t \) increases, meaning \( T(t) \) approaches \( S \).
- If \( k > 0 \), then \( T(t) \) approaches infinity, which isn't typically relevant in cooling scenarios.
In most cooling scenarios, we assume \( k < 0 \). Therefore, the horizontal asymptote represents the surrounding (ambient) temperature \( S \), indicating that as time goes on, the temperature of the object \( T(t) \) will asymptotically approach the surrounding temperature. Essentially, this means that over a long period of time, the object's temperature stabilizes at the ambient temperature, reflecting the idea that the object cools down to the temperature of its surroundings.