POLL: If you could make up ANY superhero what would he/she's name be and what would their power be?
How about this: Cut an inch of copper tubing and drill holes in it in several places on all sides. Then tape up the piece on the outside with masking tape. Then fill the tube with melted candle wax. Next attach the copper pipe to the water supply hose. Elevate the water source so you have a head. You will get exactly 0.433 pounds per square inch for each foot of elevation. You could experiment with different heights. When the fire hits the copper it will heat it (but too long of tube would take to long to heat up, so I suggested just an inch). ..and then the wax should melt and release the water. As an alternative easy solution to your problem you could just turn on the water (stop cock or clamp on rubber tubing) and put out the fire like the Parmelee sprinkler system did when it was invented in 1860. If you want to show a mathematical model of what you are doing you could calculate the density in gallons per minute per square foot of floor area. That is how sprinkler systems are designed: for a particular density. You could also determine the relationship between pressure (head) and flow, but that might require more math/physics/hydrolics than you want to deal with.
Ours beeps every 3 or 4 minutes when the battery is getting low. Pull it down and see if its one you can buy for much cheaper than your apartment offers, or just settle and call them. We couldnt afford the charge, so we actually took it down from the ceiling and left it in the car for a month lol not the best plan, but it saved our sanity.