Question

Newton's law of cooling says that the rate of cooling of an object is proportional to the difference between the temperature of the object and that of its surroundings (provided the difference is not too large). If T=T(t)T=T(t) represents the temperature of a (warm) object at time tt, AA represents the ambient (cool) temperature, and kk is a negative constant of proportionality, which equation(s) accurately characterize Newton's law?

Answer 1

Newton's law of cooling is characterized by the differential equation dT/dt = k(T - A), where T is the temperature of the object, A is the ambient temperature, and k is a negative constant of proportionality that determines the rate of cooling.

Step-by-step explanation:

Newton's law of cooling is described by a differential equation that characterizes how the temperature of a warm object, T(t), changes in relation to time, t. This law states that the rate of temperature change of the object is proportional to the difference between the object's temperature and the ambient temperature, A. The proportionality constant is denoted as k, which is a negative constant that determines the rate of cooling.

The equation that models Newton's law of cooling is expressed as:

dT/dt = k(T - A)

where dT/dt represents the rate of change of temperature with respect to time, T represents the temperature of the object at time t, A is the ambient temperature, and k is the negative constant of proportionality.

Answer 2

`T=T_o+C'e^(-kt)`

Step-by-step explanation:

Newton's law of cooling

Rate of cooling

`(dT)/(dt)\alpha(T-T_o)`

Where To is the surrounding temperature

T is object temperature at any time t

now by removing proportionality sign

`(dT)/(dt)=-k(T-T_o)`

Now by separating variables

`(dT)/((T-T_o))=-k\ dt`

`\int (dT)/((T-T_o))=-\int k\ dt`

So

`\ln (T-T_o)=- k\ t +C`

Where C is constant

`T=T_o+C'e^(-kt)`

C' is also a constant and it can be find by using boundary conditions.