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After a murder has been committed, the body, which was originally at 37°C, cools according to Newton's Law of Cooling: for some positive constant k, the rate of change of its temperature is proportional to the temperature difference with the ambient air. We will suppose that, after 1.5 hours, the temperature of the body, B, is 34.5°C, while the surrounding temperature of the ambient air is 18°C. a) Write down the differential equation governing the change of the body temperature with time, t, in terms of k b) Find the temperature B of the body as a function of k and t, the time in hours after the murder was committed and an arbitrary constant A. c) Work out the value of A and k from the information given, showing your calculation clearly.

After a murder has been committed, the body, which was originally at 37°C, cools according to Newton's Law of Cooling: for some positive constant k, the rate of change of its temperature is proportional to the temperature difference with the ambient air. We will suppose that, after 1.5 hours, the temperature of the body, B, is 34.5°C, while the surrounding temperature of the ambient air is 18°C. a) Write down the differential equation governing the change of the body temperature with time, t, in terms of k b) Find the temperature B of the body as a function of k and t, the time in hours after the murder was committed and an arbitrary constant A. c) Work out the value of A and k from the information given, showing your calculation clearly.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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After a murder has been committed, the body, which was originally at 37°C, cools according to Newton's Law of Cooling: for some positive constant k, the rate of change of its temperature is proportional to the temperature difference with the ambient air. We will suppose that, after 1.5 hours, the temperature of the body, B, is 34.5°C, while the surrounding temperature of the ambient air is 18°C. a) Write down the differential equation governing the change of the body temperature with time, t, in terms of k b) Find the temperature B of the body as a function of k and t, the time in hours after the murder was committed and an arbitrary constant A. c) Work out the value of A and k from the information given, showing your calculation clearly. d) Sketch a graph of the body temperature B against time. e) What happens to the body temperature B in the long run? Consider the limit of your solution and compare to your graph. f) Determine the time at which the murder was committed if the body is found at 16:30 at a temperature of 31°C.
Transcribed Image Text:After a murder has been committed, the body, which was originally at 37°C, cools according to Newton's Law of Cooling: for some positive constant k, the rate of change of its temperature is proportional to the temperature difference with the ambient air. We will suppose that, after 1.5 hours, the temperature of the body, B, is 34.5°C, while the surrounding temperature of the ambient air is 18°C. a) Write down the differential equation governing the change of the body temperature with time, t, in terms of k b) Find the temperature B of the body as a function of k and t, the time in hours after the murder was committed and an arbitrary constant A. c) Work out the value of A and k from the information given, showing your calculation clearly. d) Sketch a graph of the body temperature B against time. e) What happens to the body temperature B in the long run? Consider the limit of your solution and compare to your graph. f) Determine the time at which the murder was committed if the body is found at 16:30 at a temperature of 31°C.
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