Math Problem Statement
Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. Let k > 0 be the constant of proportionality. Assume the coffee has a temperature of 190 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 175 degrees in a room at 68 degrees.
(a) Write an initial value problem for the temperature T of the coffee, in Fahrenheit, at time t in minutes. Your answer will contain the unknown constant k: dT/dt = T (0) = (b) Solve the initial value problem in part (a). Your answer will contain the unknown constant k. T(t) = (c) determine the value of the constant k k = (d) Determine when the coffee reaches a temperature of 150 degrees. ______ minutes
Solution
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Exponential Decay
Newton's Law of Cooling
Formulas
dT/dt = -k(T - T_ambient)
T(t) = T_ambient + Ce^(-kt)
Theorems
Newton's Law of Cooling
Suitable Grade Level
Grades 11-12
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