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) T(t)=k(T_(a)-T) (B) (dT)/(dt)=k(T_(a)-T) (C) (dT)/(dt)=(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)  T(t)=k(T_(a)-T) (B)  (dT)/(dt)=k(T_(a)-T) (C)  (dT)/(dt)=(k)/((T_(a)-T)) (D)  T(t)=(k)/((T_(a)-T))

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Understand the problem: Understand the problem. The problem states that the rate of change of the temperature of the drink with respect to time (dT/dt) is proportional to the difference between the ambient temperature (T_a) and the current temperature of the drink (T). This is a statement of a rate of change, which is a derivative with respect to time.Translate into equation: Translate the problem into a mathematical equation. Since the rate of change of the temperature is proportional to the difference in temperatures, we can express this as an equation involving a derivative. The constant of proportionality is represented by k. The equation should therefore be of the form: (dT/dt) = k * (T_a - T)Match with options: Match the equation with the given options. We are looking for an equation that represents the rate of change of the temperature (dT/dt) being proportional to the difference (T_a - T). The correct equation must have the derivative dT/dt on one side and the expression involving the difference in temperatures on the other side, multiplied by the constant k.Identify correct option: Identify the correct option. Option (A) T(t) = k(T_a - T) does not have the derivative form we are looking for. Option (B) (dT/dt) = k(T_a - T) matches the form we derived in Step 2. Option (C) (dT/dt) = (k)/(T_a - T) has the derivative form but the relationship is not proportional, it is inversely proportional. Option (D) T(t) = (k)/(T_a - T) does not have the derivative form and also represents an inverse proportionality. The correct option is (B) because it correctly represents the rate of change of the temperature being proportional to the difference in temperatures.

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