Home » Blog » Use differential equations to find the temperature of a cooling body

# Use differential equations to find the temperature of a cooling body

A body cools from 200 degrees to 120 degrees in 30 minutes in a room at a temperature of 60 degrees.

1. Prove that the temperature of the body at time minutes is where 2. Prove that the formula for the time minutes it takes the body to cool to a temperature degrees is given by where .

3. Find the time at which the temperature of the cooling body is 90 degrees.
4. If the room temperature is falling at a constant rate of 1 degree every 10 minutes find a formula for the temperature of the cooling body at time . Assume the initial conditions are the same, the room is at 60 degrees when the body is at 200 degrees.

1. Proof. From Newton’s law of cooling (example 3 on page 315 of Apostol) we know is the unique solution to the differential equation telling us the temperature of a cooling body at time under the conditions that is the temperature of the room, and is the initial condition. In this problem we are given , . Therefore, Since we are given the additional information that , we can compute , 2. Proof. From part (a) we know So, if then 3. We compute, 4. If the temperature of the room falls at a rate of 1 degree every 10 minutes then instead of in the equation in part(a) we have . Computing as we did in part (a) with this new value for we have 