The warming or cooling rate of a drink is proportional to the difference between the ambient temperature T_(a) and the current temperature T of the drink. Which equation describes this relationship? Choose 1 answer: (A) (dT)/(dt)=(k)/((T_(a)-T)) (B) (dT)/(dt)=k(T_(a)-T) (C) T(t)=k(T_(a)-T) (D) T(t)=(k)/((T_(a)-T))

Question

The warming or cooling rate of a drink is proportional to the difference between the ambient temperature  T_(a) and the current temperature  T of the drink. Which equation describes this relationship? Choose 1 answer: (A)  (dT)/(dt)=(k)/((T_(a)-T)) (B)  (dT)/(dt)=k(T_(a)-T) (C)  T(t)=k(T_(a)-T) (D)  T(t)=(k)/((T_(a)-T))

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Understand Problem: Understand the problem and the concept of proportionality. The rate of change of the temperature of the drink with respect to time, denoted as (dT)/(dt), is said to be proportional to the difference between the ambient temperature, T_(a), and the current temperature of the drink, T. This means that as the difference between T_(a) and T changes, the rate of change of T will change in direct proportion to this difference. The constant of proportionality is denoted by k.Identify Representation: Identify the correct mathematical representation of proportionality. Proportionality in this context means that the rate of change of the temperature of the drink is equal to the constant k multiplied by the difference (T_(a) - T). Therefore, the correct mathematical representation of this relationship is a differential equation of the form: (dT)/(dt) = k * (T_(a) - T)Match Mathematical Form: Match the correct mathematical representation with the given options. Looking at the options provided, we need to find the one that matches the form identified in Step 2. Option (A) has a division by the difference, which is incorrect. Option (C) and (D) are not differential equations, as they do not represent the rate of change of T with respect to time. Option (B) is the correct form, as it represents the rate of change of T with respect to time and is directly proportional to the difference (T_(a) - T).

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