Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.


c. Why is the condition A < T₀/2 needed?

Verified step by step guidance

1. Recall that the cooling process is often modeled by Newton's Law of Cooling, which states that the rate of change of the temperature \(T(t)\) of the object is proportional to the difference between its temperature and the ambient temperature \(A\). Mathematically, this is expressed as: \[\frac{dT}{dt} = -k (T - A)\] where \(k > 0\) is a constant related to the cooling rate.
2. The solution to this differential equation with initial condition \(T(0) = T_0\) is: \[T(t) = A + (T_0 - A) e^{-k t}\] This shows that the temperature approaches the ambient temperature \(A\) as time \(t\) increases.
3. The time \(t_{1/2}\) is defined as the time when the temperature reaches half of the initial temperature, i.e., when \(T(t_{1/2}) = \frac{T_0}{2}\). Substitute this into the solution: \[\frac{T_0}{2} = A + (T_0 - A) e^{-k t_{1/2}}\]
4. To solve for \(t_{1/2}\), isolate the exponential term: \[e^{-k t_{1/2}} = \frac{\frac{T_0}{2} - A}{T_0 - A}\] For this expression to make sense (specifically, for the right side to be positive and less than 1), the numerator \(\frac{T_0}{2} - A\) must be positive. This requires that \[A < \frac{T_0}{2}\]
5. Therefore, the condition \(A < \frac{T_0}{2}\) is needed to ensure that the temperature \(\frac{T_0}{2}\) is actually between the ambient temperature \(A\) and the initial temperature \(T_0\). Without this condition, the object cannot cool down to \(\frac{T_0}{2}\) because the ambient temperature is already higher or equal to that value.

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature. This law models how objects cool or warm over time, typically leading to an exponential decay toward the ambient temperature.

Exponential Decay and Temperature Limits

The temperature of a cooling object approaches the ambient temperature asymptotically, never going below it. Understanding exponential decay helps explain why the object’s temperature decreases over time but stabilizes at the ambient temperature, which sets a lower bound for cooling.

Importance of the Condition A < T₀/2

The condition A < T₀/2 ensures that the ambient temperature is lower than half the initial temperature, making it possible for the object to cool down to T₀/2. If A were greater or equal to T₀/2, the object could never reach that temperature since it cannot cool below the ambient temperature.