Surviving the Wilderness

Launch worksheet

Launch solution 

The Scenario

You are the pilot of a firefighting plane that drops water on dangerous wilderness fires. During a long range training mission, your plane suddenly loses all powers and radio communications. You crash land in the jungle, miraculously surviving the crash without any serious injuries. Without an emergency signal to home in on, rescuers may never find you. It’s up to you to make the long trek back out of the jungle, surviving using your wits.

Math Scenario 1 – Working Out the Nearest City

You remember the approximate speed you were travelling – 200 knots, the time since you’d left your home airport – about 3 hrs, the direction of the flight – north west, and the rough arrangement of towns in the jungle:

You need to decide which city you’re going to try and reach.

You can sketch out a co-ordinate frame and calculate where you are. We can set (0, 0) to the where you started at the airport.

Firstly we can work out the (x, y) coordinates of where we have crashed. Let’s convert the speed into km/hr first:

Speed = 200 knots * 1.852 km/hr / knot

Speed = 370.4 km/hr

The (x, y) crash coordinates can be calculated using trigonometry, or using Pythagoras’ theorem:

x = -370.4 km/hr * 3 hr * cos(45)

x = -785.7 km

And:

y = 370.4 km/hr * 3 hr * sin(45) 

y = 785.7 km

So the crash site is approximately at the location (-785.7, 785.7) in the map you’ve sketched.

Now we can work out the distance from the crash site to the two cities and Pythagoras’ theorem.

Crash Site to City 1 Distance:

Distance^2 = xdist^2 + ydist^2

xdist is the distance in the horizontal, x-axis direction, and ydist is the distance in the vertical, y-axis direction

Distance^2 = (-785.7 – 0)^2 + (785.7 – 785.7)^2

Distance = 785.7 km

Crash Site to City 2 Distance:

Distance^2 = xdist^2 + ydist^2

Distance^2 = (-1000 – -785.7)^2 + (200 – 785.7)^2

Distance = 623.7 km

So the distance to City 2 is the shorter and hence best option for surival.

Math Scenario 2 – Short Hard Path or Long Easy Path

A while after setting out on your trek back to civilization, you come across a massive mountain range. You can see that the mountain range ends in the far distance so you have two choices – try and go up and over the mountain range on a much more direct path, or take the long way around, bypassing the mountains.

You sketch another map and estimate the distances involved:

50 km direct over the mountain, 25 km up and 25 km down

180 km the long way around, but flat

You estimate your speed climbing up the mountain will be about 0.5 km/hr, while your speed coming down the far side will be about 3 km/hr

You also estimate your speed on the flat is about 4 km/hr

Which route should you take? Let’s calculate the mountain route first:

Time to go over mountain = time to climb + time to descend

Time to go over mountain = 25 km / 0.5 km/hr + 25 km / 3 km/hr

Time to go over mountain = 50 hr + 8.33 hr

Time to go over mountain = 58.33 hr

And then the long way round:

Time to go long way around = distance / average speed

Time to go long way around = 180 / 4

Time to go long way around = 45 hr

So even though the long way around is more than 3 times as far, because you have a much higher average speed it’s still the quickest (and possibly safer) option.

Math Scenario 3 – Food

You have some basic wilderness survival skills and are able to set snares to catch local game and forage for edible plants and fruits. However, the longer you stay out in the wilderness, the less likely you are to make it back safely without injury or sickness, or worse.

You remember that moderate hiking burns up about 10000 kJ / day. You figure you can go into a caloric deficit for a while, but you can’t stop eating completely, and your walking speed will reduce. Each hour you spend foraging for food instead of walking nets you 1000 kJ of food.

Let’s say your average walking speed is described by this formula:

Speed = 4 km / hr – (caloric deficit / 5000)^2

You can walk for 10 hours a day, less however many hours you spend finding food. So your distance travelled in a day is:

Daily travel distance = (10 hrs – time spent foraging) * walking speed

Daily travel distance = (10 hrs – time spent foraging) * [4 km / hr – (caloric deficit / 5000)^2]

Now the caloric deficit is determined by how many hours you spend foraging:

caloric deficit = 10000 – 1000 * time spent foraging

So:

Daily travel distance = (10 hrs – time spent foraging) * [4 km / hr – ( (10000 – 1000 * time spent foraging) / 5000)^2]

Let’s represent the “time spent foraging” with the variable f:

Daily travel distance = (10 – f) * [4 – ( (10000 – 1000f) / 5000)^2]

Daily travel distance = (10 – f) * [4 – ( (2 – 0.2f))^2]

Daily travel distance = (10 – f) * [4 – (4 – 0.8f + 0.04f^2) ]

Daily travel distance = (10 – f) * [0.8f – 0.04f^2) ]

Daily travel distance = (10 – f) * [0.8f – 0.04f^2) ]

We can plot this function on a graph:

So it looks like the optimal time to spend foraging per day for the largest travel distance is just over 4 hours per day.

It’s also possible to solve this question using differentiation. First expand out the equation:

(10 – f) * [0.8f – 0.04f^2) ]

8f – 0.4f^2 – 0.8f^2 + 0.04f^3

0.04f^3 – 1.2f^2 + 8f

 

Now differentiate the distance function with respect to the foraging time f, and set it to equal zero to find the turning points in the function:

0.12f^2 – 2.4f + 8 = 0

Solve for f using the quadratic formula:

f = 15.8, 4.2

15.8 is outside the allowable foraging range, but 4.2 is within the range – and matches nicely with the answer we found using the graphical method above.

Math Scenario 4 – Rafting Downriver

In the jungle, as in the real life example, one rule of thumb is to find a river and then follow it downstream until you come across some form of civilization.

However, the river that you find has the occasional waterfall along it. You need to work out whether you’re going to be safer on the water going fast on a raft and braving the occasional waterfall, or walking alongside it going slower.

You figure you’ll need to follow the winding river for about 1000 km to get back to the nearest city. From initial scouting of the river, it looks like there are significant waterfalls approximately every 50 km. So you’ll need to brave about 20 waterfalls before getting to civilization. You think you can travel about 50 km/day using the river’s current and some paddling. With your trusty raft, you figure you have a 95% chance of getting through each waterfall unscathed.

If you choose to walk alongside the river, you can move much slower – 25 km/day, for a 40 day journey. Each night in the jungle, you think there’s a 1% chance you will get attacked and eaten by a wild animal.

Which option is the safest, and what is the chance you’ll get back to civilization safely?

Option 1: Rafting

Rafting means you need to get through 50 waterfalls. The chances of you getting through 50 waterfalls unscathed is:

Overall safety chances = (chance of getting through 1 waterfall)^(number of waterfalls)

Overall safety chances = 0.95^20

Overall safety chances = 0.3585

That’s about a 1 in 3 chance of getting back safely for the rafting option.

Option 2: Walking

Overall safety chances = (chance of surviving the night)^(number of nights)

Overall safety chances = 0.98^40

Overall safety chances = 0.4457

That’s almost a 50-50 chance of survival. So it looks like walking rather than rafting is better in this scenario.

Real Life Example – Juliane Koepcke

Juliane Koepcke was the only survivor out of 92 passengers and crew when her plane crashed into the jungle in Peru in 1971. She was only 17 years old (like a grade 12 student) when she survived the plane crash and then made an incredible journey out of the jungle.

She found a stream and went downstream (a common survival tactic), which also provided her with fresh water. After 9 days she found a shelter, treated her wounds, and waited for rescuers to find her.