Show that by Newton's Law of Cooling, the time required to...

Show that by Newton's Law of Cooling, the time required to cool
an object from temperature $A$ to temperature $B$ is
$$t=\frac{1}{k} \ln \left(\frac{A-T_{0}}{B-T_{0}}\right)$$
where $T_{0}$ is the ambient temperature.

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Key Concepts

Separation of Variables

This method is used to solve differential equations by isolating the dependent and independent variables on different sides of the equation, allowing each side to be integrated separately. It is critical for deriving the exponential decay solution in cooling problems.

Exponential Decay and Logarithms

Exponential decay describes processes where quantities decrease at a rate proportional to their current value,—as seen in the cooling process. Logarithms are used to solve the resulting exponential equations, particularly in isolating the time variable in the cooling formula.

Newton's Law of Cooling

This law states that the rate at which an object changes temperature is proportional to the difference between its current temperature and the ambient temperature. It models how an object cools or warms exponentially towards the ambient temperature over time.

First-Order Linear Differential Equations

This concept involves equations where the rate of change of a variable is expressed as a linear function of the variable itself. In the context of cooling, the temperature change is modeled by a first-order linear differential equation, making techniques like separation of variables applicable.

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