Newton's Law of cooling, \( \frac{d T}{t}=k\left(T-T_{m}\right) \) the constant \( k \) is Select the correct answer. a constant of integration evaluated from an initial condition a constant of integration evaluated from another condition a proportionality constant evaluated from an initial constant evaluated from another condition

To understand the options provided regarding the constant \( k \) in Newton's Law of Cooling, let's analyze the equation: \[ \frac{dT}{dt} = k(T - T_m) \] Here, \( T \) is the temperature of the object, \( T_m \) is the temperature of the surrounding medium, and \( k \) is a constant that characterizes the rate of cooling. ### Analysis of the Options: 1. **A constant of integration evaluated from an initial condition**: - This option suggests that \( k \) is derived from an initial condition. However, \( k \) is not a constant of integration; it is a proportionality constant that describes the rate of cooling. 2. **A constant of integration evaluated from another condition**: - Similar to the first option, this implies that \( k \) is derived from some condition, but it does not accurately describe the nature of \( k \) in this context. 3. **A proportionality constant evaluated from an initial constant evaluated from another condition**: - This option correctly identifies \( k \) as a proportionality constant. It indicates that \( k \) can be determined based on the initial conditions and possibly other conditions related to the cooling process. ### Conclusion: The most accurate description of \( k \) in the context of Newton's Law of Cooling is: **A proportionality constant evaluated from an initial constant evaluated from another condition.** Thus, the correct answer is the third option.