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A hot object is placed in a room where the ambient temperature is \(20^\circ\)C. Its temperature \(T\) decreases according to Newton’s law of cooling:$$\frac{dT}{dt} = -k(T-20),$$where \(k\) is a positive constant. At a certain moment, the object’s temperature is \(80^\circ\)C and is cooling at a rate of \(2^\circ\)C per minute.
Use the data to determine the constant \(k\).
Using Newton’s law of cooling, find the value of \(\dfrac{dT}{dt}\) when \(T = 50^\circ\text{C}\).
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