A cup of coffee with cooling constant k=0.09 min^-1 is placed in a room at temperature 20^∘ C.

step 1 The coffee and its cooling constant is 0 .09, and the temperature of the surrounding room is 20

A cup of coffee with cooling constant k=0.09 min^-1 is...

A cup of coffee with cooling constant $k=0.09 \mathrm{~min}^{-1}$ is placed in a room at temperature $20^{\circ} \mathrm{C}$. (a) How fast is the coffee cooling (in degrees per minute) when its temperature is $T=80^{\circ} \mathrm{C} ?$ (b) Use the Linear Approximation to estimate the change in temperature over the next 6 s when $T=80^{\circ} \mathrm{C}$. (c) If the coffee is served at $90^{\circ} \mathrm{C}$, how long will it take to reach an optimal drinking temperature of $65^{\circ} \mathrm{C} ?$

A cup of coffee with cooling constant $k=0.09 \mathrm{~min}^{-1}$ is placed in a room at temperature $20^{\circ} \mathrm{C}$.
(a) How fast is the coffee cooling (in degrees per minute) when its temperature is $T=80^{\circ} \mathrm{C} ?$
(b) Use the Linear Approximation to estimate the change in temperature over the next 6 s when $T=80^{\circ} \mathrm{C}$.
(c) If the coffee is served at $90^{\circ} \mathrm{C}$, how long will it take to reach an optimal drinking temperature of $65^{\circ} \mathrm{C} ?$

Key Concepts

Solving Exponential Equations with Logarithms

To determine the time required for the temperature to reach a certain value in exponential decay models, one typically isolates the exponential term and applies logarithms. This process converts the exponential equation into a linear equation in the time variable, making it easier to solve.

Newton's Law of Cooling

This principle states that the rate at which an object cools (or heats) is proportional to the difference between its current temperature and the ambient temperature. It is modeled by a first?order differential equation, often written as dT/dt = -k(T - T_ambient), where k is a positive constant that depends on the properties of the object and its environment.

Exponential Decay Model

When solving Newton's Law of Cooling, the temperature evolution over time is given by an exponential decay function. The solution, T(t) = T_ambient + (T_initial - T_ambient)e^(-kt), describes how the temperature asymptotically approaches the ambient temperature as time increases.

Derivative as a Measure of Rate of Change

The derivative in this context represents the instantaneous rate at which the temperature is changing, indicating how quickly the object is cooling or warming at a given moment. It is essential for determining the cooling speed at a specific temperature.

Linear Approximation

This is a technique used in calculus to estimate the value of a function near a given point by using the tangent line at that point. It employs the derivative to approximate small changes in temperature over a short time interval, assuming the function behaves linearly in that vicinity.

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