A group of researchers conducted an experiment to determine at what temperature does coffee tastes the best. After dozens of cups of coffee and countless sleepless nights, they have come up with an answer. It was found out that coffee tastes the best at So, let's say we have a cup of coffee which is . How long should we wait for it to turn to ? For the calculation, let's say that the coffee turned after minutes. We can solve this problem by using Newton's law of cooling. According to Newton's law of cooling, the change in temperature or the speed of cooling is proportional to difference between the object's temperature and the temperature of the surrounding medium . This can be expressed as a differential equation which goes like the following : ``` ( k is a rate constant ) ``` Let's say that the cup of coffee was in a room. With this as a condition let's solve the problem. Since then . To find , we will have to use some techniques of integration. So, But how do we find the constant ? By using the initial condition. The initial condition is that the coffee is in the beginning. So : Therefore What we are looking for is the time it takes for the coffee to become . Hence we should set and solve for . and therefore Here we are stuck once again. How do we find the rate constant ? For this, we use the condition which says that the coffee turns after minutes. So : turns to Finally we get an equation : Since we want to find when is : minutes The morale of the story -- drink coffee minutes after you made it.