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Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between the temperature of the object itself and the temperature of its surroundings (also known as the ambient temperature). As a simplifying assumption, we may assume that for many cases, this ambient temperature, Ta, is a constant. Suppose for a particular experiment that the ambient temperature is 70°F and that the rate constant of proportion- ality is known to be 0.05min ¹. (a) Write and solve a differential equation which gives the temperature of the object at any time. (Note that, in the absence of any initial conditions, your solution should be the general solution of the ODE.) (b) Suppose that you were setting up an experiment using the above information, but you forgot to note down the starting temperature of the object (when t = 0). After 5 minutes, you measured the temperature of the object to be 120°F. What was the starting temperature?

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between the temperature of the object itself and the temperature of its surroundings (also known as the ambient temperature). As a simplifying assumption, we may assume that for many cases, this ambient temperature, Ta, is a constant. Suppose for a particular experiment that the ambient temperature is 70°F and that the rate constant of proportion- ality is known to be 0.05min ¹. (a) Write and solve a differential equation which gives the temperature of the object at any time. (Note that, in the absence of any initial conditions, your solution should be the general solution of the ODE.) (b) Suppose that you were setting up an experiment using the above information, but you forgot to note down the starting temperature of the object (when t = 0). After 5 minutes, you measured the temperature of the object to be 120°F. What was the starting temperature?

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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Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between the temperature of the object itself and the temperature of its surroundings (also known as the ambient temperature). As a simplifying assumption, we may assume that for many cases, this ambient temperature, Ta, is a constant. Suppose for a particular experiment that the ambient temperature is 70°F and that the rate constant of proportion- ality is known to be 0.05min ¹. (a) Write and solve a differential equation which gives the temperature of the object at any time. (Note that, in the absence of any initial conditions, your solution should be the general solution of the ODE.) (b) Suppose that you were setting up an experiment using the above information, but you forgot to note down the starting temperature of the object (when t = 0). After 5 minutes, you measured the temperature of the object to be 120°F. What was the starting temperature?
Transcribed Image Text:Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between the temperature of the object itself and the temperature of its surroundings (also known as the ambient temperature). As a simplifying assumption, we may assume that for many cases, this ambient temperature, Ta, is a constant. Suppose for a particular experiment that the ambient temperature is 70°F and that the rate constant of proportion- ality is known to be 0.05min ¹. (a) Write and solve a differential equation which gives the temperature of the object at any time. (Note that, in the absence of any initial conditions, your solution should be the general solution of the ODE.) (b) Suppose that you were setting up an experiment using the above information, but you forgot to note down the starting temperature of the object (when t = 0). After 5 minutes, you measured the temperature of the object to be 120°F. What was the starting temperature?
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