Question

find a mathematical model for the verbal statement. (use k for the constant of proportionality.) the rate of change r of the temperature of an object is directly proportional to the difference between the temperature t of the object and the temperature t_e of the environment.

Explanation:

Step1: Recall direct - proportionality concept

Direct proportionality means if one quantity $y$ is directly proportional to another quantity $x$, then $y = kx$, where $k$ is the constant of proportionality.

Step2: Identify the two related quantities

Here, the rate of change $R$ of the temperature of an object is the dependent quantity and the difference between the temperature $T$ of the object and the temperature $T_e$ of the environment, i.e., $(T - T_e)$ is the independent quantity.

Step3: Write the mathematical model

Based on the direct - proportionality formula, we can write $R=k(T - T_e)$.

Answer:

$R = k(T - T_e)$