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.


a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

Verified step by step guidance

1. Start with Newton's Law of Cooling, which states that the temperature \( T(t) \) of the object at time \( t \) satisfies the differential equation: \[\frac{dT}{dt} = -k (T - A)\] where \( k > 0 \) is a constant and \( A \) is the ambient temperature.
2. Solve the differential equation by separating variables or recognizing it as a first-order linear ODE. The general solution is: \[T(t) = A + (T_0 - A) e^{-k t}\] where \( T_0 \) is the initial temperature at \( t = 0 \).
3. Define \( t_{1/2} \) 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. Isolate the exponential term: \[\frac{T_0}{2} - A = (T_0 - A) e^{-k t_{1/2}}\] Then divide both sides by \( T_0 - A \): \[\frac{\frac{T_0}{2} - A}{T_0 - A} = e^{-k t_{1/2}}\]
5. Take the natural logarithm of both sides to solve for \( t_{1/2} \): \[-k t_{1/2} = \ln \left( \frac{\frac{T_0}{2} - A}{T_0 - A} \right)\] Rearranging gives: \[t_{1/2} = -\frac{1}{k} \ln \left( \frac{T_0 - 2A}{2 (T_0 - A)} \right)\] which is the desired expression.

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. Mathematically, this is expressed as dT/dt = -k(T - A), where k > 0 is a constant. This law models how objects cool or warm over time toward the surrounding temperature.

Solving First-Order Linear Differential Equations

The temperature function T(t) satisfies a first-order linear differential equation. Solving it involves separating variables or using an integrating factor to find T as a function of time. This solution typically involves an exponential decay term reflecting how temperature approaches ambient temperature over time.

Logarithmic Manipulation in Solving for Time

To find the cooling time t₁/₂ when the temperature reaches T₀/2, one must solve the exponential equation for t. This requires isolating t by taking the natural logarithm of both sides, using properties of logarithms to simplify the expression, and expressing t₁/₂ in terms of known quantities like T₀, A, and k.