(1 point) Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature andthat of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 185 degreesFahrenheit when freshly poured, and 2.5 minutes later has cooled to 173 degrees in a room at 74 degrees, determine when the coffee reaches atemperature of 123 degrees.The coffee will reach a temperature of 123 degrees inminutes.

Answered: (1 point) Newton's law of cooling…

Linear Algebra: A Modern Introduction

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Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

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(1 point) Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and
that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 185 degrees
Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 173 degrees in a room at 74 degrees, determine when the coffee reaches a
temperature of 123 degrees.
The coffee will reach a temperature of 123 degrees in
minutes.

Expert Solution

Step 1

Let $T (t)$ be the temperature of coffee in $t$ minutes.

Let $T_{s}$ be the surrounding temperature. It is given that room temperature is $74$ degrees. Hence,

$T_{s} = 74$

The rate of change of the function $T (t)$ is $\frac{d T}{d t}$ .

The temperature of an object changes at a rate proportional to the difference between its temperature and surrounding temperature.

$\frac{d T}{d t} α (T - T_{s})$

Hence, there exists a constant $k$ such that,

$\frac{d T}{d t} = k (T - T_{s})$

Substitute $T_{s} = 74$ :

$\frac{d T}{d t} = k (T - 74)$

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