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A cup of hot coffee initially at 2050 F cools to 165° F in 4 min while sitting in a restaurant of temperature 70° F. When the temperature of the coffee will be 1200 F

A cup of hot coffee initially at 2050 F cools to 165° F in 4 min while sitting in a restaurant of temperature 70° F. When the temperature of the coffee will be 1200 F

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
**Problem Statement:** A cup of hot coffee initially at 205°F cools to 165°F in 4 minutes while sitting in a restaurant with a temperature of 70°F. When will the temperature of the coffee be 120°F? **Explanation:** This problem involves cooling as the coffee loses heat to the surrounding environment (the restaurant). The scenario is based on Newton's Law of Cooling, which describes the rate of heat transfer between a warmer object and a cooler environment. You need to determine the time it takes for the coffee to reach a temperature of 120°F.
Transcribed Image Text:**Problem Statement:** A cup of hot coffee initially at 205°F cools to 165°F in 4 minutes while sitting in a restaurant with a temperature of 70°F. When will the temperature of the coffee be 120°F? **Explanation:** This problem involves cooling as the coffee loses heat to the surrounding environment (the restaurant). The scenario is based on Newton's Law of Cooling, which describes the rate of heat transfer between a warmer object and a cooler environment. You need to determine the time it takes for the coffee to reach a temperature of 120°F.
Expert Solution
Step 1: Newton's Law of Cooling

According to Newton’s law of cooling, the rate of change of temperature of a body is directly proportional to the difference in the temperature of the body and its surroundings.

i.e. 

table row cell fraction numerator d T over denominator d t end fraction end cell proportional to cell open parentheses T minus T subscript s close parentheses end cell row blank rightwards double arrow cell fraction numerator d T over denominator d t end fraction equals k open parentheses T minus T subscript s close parentheses end cell end table

Where,

  • T= Temperature of the body at time t
  • Ts = Temperature of the surrounding
  • k = proportionality constant.
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