1. Newton’s Law of Cooling

2. History
In the 17th century, Isaac Newton studied the nature of cooling
In his studies he found that if there is a less than 10 degree difference between two objects the rate of heat loss is proportional to the temperature difference
Newton applied this principle to estimate the temperature of a red-hot iron ball by observing the time which it took to cool from a red heat to a known temperature, and comparing this with the time taken to cool through a known range at ordinary temperatures.
According to this law, if the excess of the temperature of the body above its surroundings is observed at equal intervals of time, the observed values will form a geometrical progression with a common ratio
However, Newton’s law was inaccurate at high temperatures
Pierre Dulong and Alexis Petit corrected Newton’s law by clarifying the effect of the temperature of the surroundings

3. What is it?
Newton's Law of Cooling is used to model the temperature change of an object of some temperature placed in an environment of a different temperature. The law states that:
y = the temperature of the object at time t
T0 = the temperature of the surrounding environment (constant)
k = the constant of proportionality
This law says that the rate of change of temperature is proportional to the difference between the temperature of the object and that of the surrounding environment.

4. The Basic Concept
In order to get the previous equation to something that we can use, we must solve the differential equation. The steps are given below.
Separate the variables. Get all the y's on one side and the t on the other side. The constants can be on either side.
Integrate each side
Find antiderivative of each side
Leave in the previous form or solve for T
We now have a useful equation. When you are working with Newton's Law of Cooling, remember that t is the variable. The other letters, R, k, C, are all constants. In order to find the temperature of the object at a given time, all of the constants must first have numerical values.

5. The Problem
Bob and Kate are cranking out math problems at their home. Fred is at his home making pizza. He calls Bob and tells him that he is taking the pizza out from the oven right now. Bob and Kate need to get back home in time so that they can enjoy the pizza at a warm temperature of 43°C. The pizza, heated to a temperature of 230°C, is taken out of an oven and placed in a 24°C room at time t=0 minutes. If the pizza cools from 230° to 190° in 1 minute, how much longer will it take for its temperature to decrease to 43°?

6. How to do it It takes about 11.04 more minutes
For the object to cool to a temperature of
43°

7. Problem 2
If a metal container of hot water at 700C cools to 400C in 5 minutes in a room at 200C, calculate its temperature after 10 minutes

8. Real Life Applications
To predict how long it takes for a hot cup of tea to cool down to a certain temperature
To find the temperature of a soda placed in a refrigerator by a certain amount of time.
In crime scenes, Newton’s law of cooling can indicate the time of death given the probable body temperature at time of death and the current body temperature