Flashback to Differential Equations

My day in a nutshell:

7:30am- Burning smell in house, thermostat reads 52F

9:45am- Repairman shows up, determines furnace is broken, calls someone else to fix it

1:30pm- Indoor temp=47F

4:45pm- 2 new repairmen lug furnace motor off to their shop

6:30pm- Motor returns, furnace works, temp=43F

For an hour or two it sounded like the repair wouldn't be finished until tomorrow morning. That would have made for a miserable night as the inside temperature would have dropped even faster because of the forecasted low of 11 below zero tonight. 

That got me wondering... 

If they hadn't fixed it tonight, how cold would it have been inside by the time they fixed it tomorrow? I tackled this problem with differential equations. (These are equations about rates of change.) I applied Newton's Law of Cooling and assumed that the change in the temperature of my house is proportional to the difference between its temp and the outdoor temp. 

I worked things down to this equation:

, where T(t) is the temperature of the house at time t, Ta is the outside temperature, T0 is the initial inside temperature, and k is a constant. To make things simpler I assumed a constant outdoor temperature of 8F. Knowing the inside temp at two different times throughout the day made it easy to solve for k and complete the equation.

According to this equation, if the repair had waited until 9am tomorrow, the temperature inside would have been approximately 34.38F. This is an extremely conservative calculation as it assumes the outdoor temperature is constant, when in actuality it will drop considerably.  The temperature in the house would have dropped below freezing during the night if the furnace had not been repaired. 

To have even more fun with this math problem, I decided to graph it :)  Keep in mind the equation is conservative and the temperatures would have actually been even lower. 

As fun as it might have been to compare the predicted and actual temps, I am stoked to have heat again!