Answered: Suppose that a cup of soup cooled from…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

Suppose that a cup of soup cooled from 90°C to 60°C after 25 minutes in a room whose temperature was 20°C. Use Newton's Law of Cooling to answer the following questions.

a) How much longer would it take the soup to cool to 35°C? round the final answer to 2 decimal places. Round all intermediate values to five decimal places.

Expert Solution

Step 1: Newton's Law of Cooling

Newton's Law of Cooling is represented by the following differential equation:

$dT over dt equals negative straight k left parenthesis straight T minus straight T subscript straight a right parenthesis$

$Solving the differential equation we have,$

$straight T left parenthesis straight t right parenthesis equals straight T subscript straight a plus left parenthesis straight T subscript 0 minus straight T subscript straight a right parenthesis space straight e to the power of negative kt end exponent$

Where:

$T (t)$ is the temperature of the soup at time $t$ .
$T_{a}$ is the ambient temperature (room temperature), which is 20°C.
$T_{0}$ is the initial temperature of the soup, which is 90°C.
$k$ is the cooling constant that we need to find.
$t$ is the time in minutes.

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