In Exercise 28 in Section 9.2 we discussed a differential...

In Exercise 28 in Section 9.2 we discussed a differential equation that models the temperature of a $95 ^ { \circ } \mathrm { C }$ of coffee in a $20 ^ { \circ } \mathrm { C }$ room. Solve the differential equation to find an expression for the temperature of the coffee at time $t .$

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Key Concepts

Exponential Decay

Exponential decay describes a process in which the rate of decrease of a quantity is proportional to its current value, resulting in a solution that involves an exponential function, which is a common outcome when solving cooling or decay problems.

Newton's Law of Cooling

This principle states that the rate at which an object's temperature changes is proportional to the difference between its temperature and the ambient temperature, leading to an exponential model of temperature change over time.

First Order Linear Differential Equations

These are equations that involve the first derivative of a function and the function itself, and they can often be solved using techniques like separation of variables or integrating factors, providing a framework for modeling various natural phenomena such as cooling or warming.

Initial Value Problem

An initial value problem involves solving a differential equation subject to a specific starting condition or initial state, which allows for the determination of particular solutions that describe the system’s behavior from the given starting point.

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