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Newton’s Law of Cooling. According to Newton’s law of
Chapter 2, Problem 34E(choose chapter or problem)
Newton’s Law of Cooling. According to Newton’s law of cooling, if an object at temperature T is immersed in a medium having the constant temperature M, then the rate of change of T is proportional to the difference of temperature M - T. This gives the differential equationdT / dt = k(M – T) .(a) Solve the differential equation for T. (b) A thermometer reading 1000F is placed in a medium having a constant temperature of 700F. After 6 min, the thermometer reads 800F. What is the reading after 20 min? (Further applications of Newton’s law of cooling appear in Section 3.3.)
Questions & Answers
QUESTION:
Newton’s Law of Cooling. According to Newton’s law of cooling, if an object at temperature T is immersed in a medium having the constant temperature M, then the rate of change of T is proportional to the difference of temperature M - T. This gives the differential equationdT / dt = k(M – T) .(a) Solve the differential equation for T. (b) A thermometer reading 1000F is placed in a medium having a constant temperature of 700F. After 6 min, the thermometer reads 800F. What is the reading after 20 min? (Further applications of Newton’s law of cooling appear in Section 3.3.)
ANSWER:Solution:Step-1:a)In this problem we need to find the solution of the differential equation for T.The given differential equation is: .Here , M is the constant temperature , T is the temperature and k is the proportionality constant.Step-2:By using variable, separable method the differential equation can be written as:Integrate on both sides we get:, since - (-1) = 1, since ,here M is constant., since . , since is any constant.Step-3:Therefore, t