A body cools from 200 degrees to 120 degrees in 30 minutes in a room at a temperature of 60 degrees.
- Prove that the temperature of the body at time minutes is
where
- Prove that the formula for the time minutes it takes the body to cool to a temperature degrees is given by
where .
- Find the time at which the temperature of the cooling body is 90 degrees.
- 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.
- 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 ,
- Proof. From part (a) we know
So, if then
- We compute,
- 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