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

Answer:

$T=T_o+C'e^{-kt}$

Explanation:

Newton's law of cooling

Rate of cooling

$\dfrac{dT}{dt}\alpha(T-T_o)$

Where To is the surrounding temperature

T is object temperature at any time t

now by removing proportionality sign

$\dfrac{dT}{dt}=-k(T-T_o)$

Now by separating variables

$\dfrac{dT}{(T-T_o)}=-k\ dt$

$\int \dfrac{dT}{(T-T_o)}=-\int k\ dt$

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.