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