Question 1. Newtons law of cooling states that the rate of cooling of an object

Question

Question 1.

Newtons law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. A piece of metal at temperature of 72 (degrees) is placed in a big container where the temperature is always 44 (degrees). After half an hour, the metal has cooled to 61 (degrees)

(A) what is the temperature of the metal after another half hour?

(B) How long does it take for the metal to cool to 50 (degrees)

Please show all working.

Explanation / Answer

Solution

Let T be the temperature of the object. Then, rate of cooling = - (dT/dt).

If d represents the difference between the temperature of the object and surroundings, then the given condition => - (dT/dt) = kd, where k is a constant.

Or dT = - kddt

Integrating both sides, T =(kd)t + c, where c = constant of integration.

For the given example, T = 72 when t = 0. So substituting in the above equation, c = 72.

Further, for t = ½ hour, d = 72 – 44 = 28 and T = 61. Again substituting in the eabove equation, 61 = 72 – k(28 x ½) or k = 11/14.

Thus, the equation becomes: T = 72 - (11/14)d.

Part (A)

Another half an hour => 1 hour from the initial state. So, substituting t = 1 in the equation, we get:

T = 72 - 28(11/14) = 50 ANSWER

Part (B)

As seen in Part (A), after 1 hour from start the temperature of the object is 50. So, answer is 1hour

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