Question

According to Newton's law of cooling, the rate at which an object's temperature changes is directly proportional to the difference in temperature between the object and the surrounding medium. If T(t) represents the temperature of the object (CC) at time t (in hours), and T5 represents the constant temperature of the surrounding medium, then the differential equation best describing the rate of change in the temperature of the object is:

Answer 1

dT/dt = k[T5 - T]

Step-by-step explanation:

DT/dt represents rate of change in temperature for Celsius degrees per hour. Its proportional to the difference in temperature between the object and the Surrounding medium. This means either dT/Dt =k (T-TS) or dT/dt=k(TS-T) with k being some positive constant of proportionality which depends on the object.

We can see that dT/dt = k(T-T5) has temperature increasing when the temperature of the object T is greater than the surrounding medium (T5).

The equation dT/dt=k(T5-T) has the temperature increasing when the object (T) is less than the temperature of the surrounding medium. Therefore the differential equation best describing the rate of change in temperature of the object is dT/dt=k(T5-T) for some positive constant of proportionality k.

Answer 2

dT(t)/dt = k[T5 - T(t)]

Step-by-step explanation:

Since T(t) represents the temperature of the object and T5 represents the temperature of the surroundings, according to Newton's law of cooling, the rate at which an object's temperature changes is directly proportional to the difference in temperature between the object and the surrounding medium, that is dT(t)/dt ∝ T5 - T(t)

Introducing the constant of proportionality

dT(t)/dt = k[T5 - T(t)]

which is the desired differential equation