Firefighter

Launch worksheet

Launch solution 

The Scenario

You are the new rookie firefighter in a large town and you’re just about to be tested by the worst week in history. You’ll have to use your wits to get through a range of incredible challenges, save as many people as you can and come through safely yourself as well. Ready? Let’s go.

Math Scenario 1: Apartment Block Fire

First day on the job, a massive apartment block goes up in flames. Your fire truck is dispatched to help put out the blaze. 

Your fire truck has a capacity of 3000 litres of water on-board. Due to the debris, you’re only able to park the truck 100 metres away from the optimal location for fighting the fire.

You’ll have to wind out 120 metres of hose to get to the site.

You need 100 psi (pounds per square inch) pressure at the nozzle of the hose to operate it properly. Friction from the water passing through the hose means you need about 0.8 psi per metre of hose. Your truck can provide 200 psi at the truck end. Will you have enough pressure at the hose?

Nozzle pressure = truck pressure – pressure required for hose length

Nozzle pressure = 200 – 120 * 0.8

Nozzle pressure = 200 – 120 * 0.8

Nozzle pressure = 104 psi

You’ll have just enough pressure to properly fight the fire with that length of hose.

Math Scenario 2: Rural Blaze

Fourth day on the job, you’re called out to a growing blaze on the outskirts of town. Problem is, there’s no local water supply so you’re going to have to organize a regular ferry service.

It’s 5 km to the nearest water source, and your trucks have a top speed of 40 km/hr on the rough rural roads. Fill up time is 10 minutes.

Each truck will expend its water supply in 10 minutes when on site.

How many trucks will you need in rotation to always have one on site, fighting the fire?

To answer this question, we can work out the “duty cycle” of one truck – how long it will spend on site, in transit, filling up its tank and then back in transit to the fire site:

The total time for a complete cycle is:

Cycle time =  time on station + time in transit + time refilling + time in transit

Cycle time =  time on station + distance / speed + time refilling + distance / speed

Cycle time =  10 minutes + 5 km / 40 km/hr * 60 min/hr + 10 minutes + time in transit

Cycle time =  10 minutes + 7.5 minutes + 10 minutes + 7.5 minutes

Cycle time =  35 minutes

So one fire truck will be able to spend 10 minutes on station out of every 35 minutes.

That means you’ll need at least 3 other trucks to always have one truck on station fighting the fire.

Math Scenario 3: Freak Tornado

On Sunday, which was supposed to be your day off, you get called in to work to deal with a major disaster.

A freak tornado has ripped through the city central business district, decimating the skyscrapers and starting fires all over the place. A bit like the scene from The Day After Tomorrow:

All power and reserve power is out, so you’ll have to use the stairs.

There are many buildings in trouble, with an average height of 80 stories – that’s 80 flights of stairs. 

Your fire chief has reported that another tornado is forming, and is predicted to hit the city in about 45 minutes. You have plenty of firefighters to split between the buildings but they have to get up and back out again in time.

If it takes about 15 seconds to climb or descend each floor and 2 minutes to clear the floor of people (at each floor you drop off firefighters to do the job), how long will it take you to clear the entire building?

The trick to working this out is to realise that the longest duration will be for the firefighters that go straight to the top floor:

Total time = ascent time + clear top floor time + descend time

Total time = #floors * time to climb/descend each floor + clear top floor time + #floors * time to climb/descend each floor

Total time = 80 * 15 + 2 minutes * 60 s/min + 80 * 15

Total time = 1200 + 120 + 1200

Total time = 2520 seconds

Total time = 42 minutes

Looks like you’ll just have enough time to clear everyone out – better hurry!

Real Life Example – Emergency Workers at 9/11

In the immediate aftermath of the 9/11 attacks in New York city, emergency workers performed a heroic effort to rescue as many people as they could, at incredible risk to themselves. 411 emergency workers were killed.

Some of rescue efforts and evacuations could use the elevators, and some of it had to be done on foot using the stairs. Many people had to be rescued out of elevators.

Problems with faulty radios and and the radio repeater system in the World Trade Centre meant that, even when the order to evacuate was given after the South Tower collapsed at 9:59 am, many did not get the message.

Without the heroic efforts of these emergency responders, many hundreds more likely would have died.

Mathematical modelling of how to optimise evacuation processes during a disaster is a complex and challenging problem, which has been the focus of much research. In large buildings, many measures have been introduced including the use of lifts in some situations for evacuation (traditionally a no-no and the subject of standards) and also safe shelter areas in very large buildings.