# Outside Temp Differential Equation

• Feb 23rd 2009, 09:13 AM
leebatt
Outside Temp Differential Equation
The temperature in a room is 70 degrees F. A thermometer which has been kept in the room is placed outside. In 5 minutes the thermometer reading is 60 degrees F. Five minutes after that, it is 55 degrees F.
Find the outdoor temperature.

Thank you.
• Feb 23rd 2009, 09:30 AM
HallsofIvy
Quote:

Originally Posted by leebatt
The temperature in a room is 70 degrees F. A thermometer which has been kept in the room is placed outside. In 5 minutes the thermometer reading is 60 degrees F. Five minutes after that, it is 55 degrees F.
Find the outdoor temperature.

Thank you.

Newton's law of cooling: the rate at which heat flows from a warmer body to a cooler one is proportional to the difference in temperatures. The assumption here, of course, is that the "outdoors" is big enough that heat from the room won't change its temperature. If we let T be the temperature in the room and \$\displaystyle T_0\$ the constant outdoor temperature, then [tex]dT/dt= k(T- T_0)[tex]. Integrate \$\displaystyle dT/(T-T_0)= kdt\$. The general solution will involve a constant of integration, k, and \$\displaystyle T_0\$. The three given values, T(0)= 70, T(5)= 60, T(10)= 55, give you three equations to solve for those constants.