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Chapter 3: Problem 35

Mr. Jones puts two cups of coffee on the breakfast table at the same time. He immediately pours cream into his coffee from a pitcher that was sitting on the table for a long time. He then reads the morning paper for 5 minutes before taking his first sip. Mrs. Jones arrives at the table 5 minutes after the cups were set down, adds cream to her coffee, and takes a sip. Assume that Mr, and Mrs. Jones add exactly the same amount of cream to their cups. Discuss who drinks the hotter cup of coffee. Defend your assertion with sound mathematics.

Short Answer

Expert verified
Mr. Jones drinks the hotter coffee because he adds cream immediately, reducing heat loss compared to Mrs. Jones.

Step by step solution

01

Understanding Heat Transfer

Determine how cream affects the temperature of the coffee. Cream, initially at a lower temperature than hot coffee, cools it down. For heat transfer, coffee cools at a constant rate proportional to the temperature difference between the coffee and its surroundings due to Newton's Law of Cooling.
02

Analyze Mr. Jones's Coffee

Mr. Jones adds cream immediately, which reduces his coffee's temperature by a fixed amount. The cooled coffee then loses heat to the environment over the next 5 minutes according to Newton's Law of Cooling, which states that the rate of cooling is proportional to the temperature difference with the room temperature.
03

Analyze Mrs. Jones's Coffee

Mrs. Jones adds cream 5 minutes later. During this time, her coffee cools in the air. When she adds cream, it lowers her coffee's temperature further. However, her coffee had a greater temperature drop during the initial 5 minutes compared to Mr. Jones.
04

Apply Newton's Law of Cooling

Newton's Law of Cooling can be simplified as: \( T(t) = T_{ ext{ambient}} + (T_0 - T_{ ext{ambient}}) imes e^{-kt} \), where \( T(t) \) is the temperature at time \( t \), \( T_{ ext{ambient}} \) is ambient temperature, \( T_0 \) is initial temperature, and \( k \) is a cooling constant. Calculating for 5 minutes at different stages shows Mrs. Jones's coffee is cooler.
05

Compare Final Temperatures

Mr. Jones's coffee was cooled by cream immediately and lost heat over the 5 minutes. Mrs. Jones's coffee lost more heat initially, so the cream had a cooler coffee to be added into, resulting in a lower final temperature. Hence, Mr. Jones drinks the hotter coffee.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Transfer
Heat transfer is the movement of thermal energy from one object to another. In the scenario involving Mr. and Mrs. Jones, the coffee's initial temperature is reduced by adding cream. When cream, which is cooler than the coffee, is added, thermal energy transfers from the coffee to the cream, leading to a drop in temperature.
This process continues with the cooling of the coffee to the ambient room temperature. According to Newton's Law of Cooling, this heat transfer occurs at a rate proportional to the temperature difference between the coffee and its surroundings.
  • The greater the temperature difference, the faster the rate of heat transfer.
  • This means if you leave hot coffee sitting out, it will initially cool rapidly and then slowly as it approaches room temperature.
Understanding heat transfer is crucial in predicting who drinks the hotter coffee.
Temperature Difference
The concept of temperature difference is pivotal in determining the rate of cooling according to Newton's Law of Cooling. Mr. Jones adds cream right away, which instantly reduces the temperature difference between his coffee and the room.
This reduced difference causes his coffee to cool more slowly over time. On the other hand, Mrs. Jones waits 5 minutes before adding cream, during which time her coffee experiences a larger temperature difference with the ambient air.
  • The larger this difference, the more rapid the cooling effect.
  • This means Mrs. Jones's coffee cools faster during those 5 minutes than Mr. Jones's initially does.
While the cream further lowers her coffee's temperature, the initial faster cooling makes her coffee ultimately cooler.
Cooling Constant
The cooling constant, denoted as \( k \) in Newton's Law of Cooling, is a unique value that determines how quickly an object cools down. This constant depends on factors like the nature of the coffee cup and environmental conditions such as air flow and room temperature.
In the mathematical model: \[ T(t) = T_{\text{ambient}} + (T_0 - T_{\text{ambient}}) \times e^{-kt} \] The cooling constant \( k \) influences how rapidly the temperature approaches the ambient temperature.
  • A larger \( k \) signifies faster cooling.
  • This means environmental factors affect the cooling rate more significantly.
For Mr. and Mrs. Jones, even a slight difference in cooling constants could lead to one coffee cooling faster, further emphasizing the complexity in predicting final temperatures without precise information on \( k \).

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Most popular questions from this chapter

(a) If a constant number \(h\) of animals is removed or harvested per unit time, then a model for the population \(P(t)\) of animals at any time \(t\) is given by $$ \frac{d P}{d t}=P(a-b P)-h, \quad P(0)=P_{0}, $$ where \(a, b, h\), and \(P_{0}\) are positive constants. Solve the problem when \(a=5, b=1\), and \(h=4\). (b) Use an ODE solver to determine the long-term behavior of the population in part (a) in the cases \(P_{0}>4,1

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