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------------------------------------------------ A covered mug of coffee, originally at 220 degrees Fahrenheit, is left for \( t \) hours in a room where the temperature is 70 degrees. It will cool to a temperature of \( 70 + 150e^{-1.7t} \) degrees.

Find the temperature after \( t \) hours.

The temperature of a coffee in a mug originally at 220 degrees Fahrenheit and cooled in a room whose temperature is 70 degrees, is calculated by substituting the time factor in the given exponential decay equation.

Explanation:

The formula given represents a decaying exponential that relates the temperature of a coffee mug initially at a high temperature, to its temperature at some expected time. The equation is derived based on Newtown's Law of Cooling, and the simple physical concept that heat flows spontaneously from a hotter object to a colder object. Thus, the coffee will lose heat to its surroundings (which are at a lower temperature) over time.

To find the temperature at a specific time, you simply need to substitute the time 't' in hours into the equation. For example, to find the temperature of the coffee after one hour, replace 't' with '1' in the equation to get 70 + 150e−1.7 which equals approximately 81.74 degrees Fahrenheit.

For a perfect understanding of the physical context, consider a scenario where a spoon is placed in the hot coffee. The heat will spontaneously flow from the hotter object (coffee) to the colder object (spoon), resulting in a more uniform distribution of thermal energy as the spoon warms and the coffee cools. Understanding this concept can help better grasp why the temperature of the coffee decreases over time.

Learn more about Heat Transfer and Cooling here:

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