The cooling rate (rate of heat loss) of an object is directly proportional to the temperature difference between the body and its surroundings, provided the temperature difference is not too large (temperature difference should be less than 30°) is known as the Newton's cooling law. The mathematical representation of this law is given below:

$\frac{dT}{dt}$$\propto (T_t - T_s)$

$\frac{dT}{dt}$$ = k(T_t - T_s)$

where,

$T_t$ = temperature of a body at time t,

$T_s$ = temperature of the surrounding,

k = constant of proportionality.

**Step 1 :**Identify the data given in the question.

**Step 2 :**Plug these given parameters into the newton's law of cooling formula below and find the unknown parameter.

T(t) = $T_{s}$ + $(T_{0 }- {T_s})$$e^{-kt}$

where,

T(t) = temperature of a body at time t,

$T_s$ = temperature of the surrounding,

$T_0$ = initial temperature of the body,

k = constant of proportionality.

Below are some of the problems based on newton's law of cooling.### Solved Examples

**Question 1:**Stefen checks that the temperature inside his house is $50^{\circ}$ F. After 20m what would be the temperature if $20^{\circ}$ F is the surrounding temperature?(k value is given as 0.095.)

**Solution:**

Step 1 : Given that,

Initial temperature, $T_{0}$ = 50°F

Surrounding temperature, $T_{s}$ = 20°F

Time, t = 20m = 120s

k = 0.095

Step 2 : Formula is given by,

T(t) = $T_{s}$ + $(T_{0} - T_{s})$ $e^{-kt}$

T(t) = 20 + $(50-20)$ $e^{-0.095\times120}$ = 20°F

**Question 2:**Initial temperature of a body is 40°F and the surrounding temperature is 25°F. Calculate the temperature of a body after 30minute if k = 0.01?

**Solution:**

Step 1 : Given that,

Initial temperature, $T_{0}$ = 40°F

Surrounding temperature, $T_{s}$ = 25°F

Time, t = 30m = 180s

k = 0.01

Step 2 : Formula is given by,

T(t) = $T_{s}$ + $(T_{0} - T_{s})$ $e^{-kt}$

T(t) = 25 + $(40-25)$ $e^{-0.01\times180}$ = 27.47°F