# Activity: A Walk in the Desert

## Crash!

Jade has crash-landed in the desert.

There is a village somewhere nearby, direction unknown.

So Jade comes up with a cunning plan:

- Fill up a water bottle from the plane, and take a compass,
- Then walk 1 km north, change direction and walk 2 km east, then 3 km south, 4 km west, 5 km north, 6 km east, and so on, like this:

This way Jade will find the village no matter what direction it is in, and can (hopefully) find the way back to the plane for fresh water and shade when needed.

But Jade needs to know, at the end of each stage:

- The total distance walked
- How far (in a straight line) back to the plane

## OK, Let's Do the Calcs ...

After one stage of the journey, Jade has reached point A:

- Jade has walked 1 km altogether.
- And is 1 km (in a straight line) from the plane.

After two stages, Jade has reached point B:

- Jade has walked 3 km altogether.
- To answer the second question, we make a right-angled triangle OAB:

We can calculate the length of OB using Pythagoras' Theorem, as follows:

^{2}= OA

^{2}+ AB

^{2}

^{2}= 1

^{2}+ 2

^{2}

^{2}= 1 + 4

^{2}= 5

So the answer in this case is:

At Point B, the distance back to the plane (in a straight line) is **√5 km**

After three stages, Jade is at point C:

Hey, it's your turn Now!

Fill in all the other values ... if it gets hard, read below for some help

Point | Distance walked altogether |
Distance (in a straight line) from O |
---|---|---|

O | 0 | 0 |

A | 1 | 1 |

B | 3 | √5 |

C | 6 | |

D | ||

E | ||

F | ||

G | ||

H | ||

I | ||

J |

## How To Make It Easier

### Distance walked altogether

At the end of each stage, the total distance is the sum of the series 1 + 2 + 3 + 4 + 5 + 6 + ...

So just add the new distance each time.

**OR** you can calculate each value using:

n(n + 1)/2

where **n** is the number of stages.

Like this:

Number
of stages (n) |
Total
distance walked = n(n + 1)/2 |

1 | 1 × 2 / 2 = 1 |

2 | 2 × 3
/ 2 = 3 |

3 | 3 × 4
/ 2 = 6 |

4 | etc ... |

This works because it is the "Triangular Number Sequence":

Try doing the calculations both ways, for fun.

### Distance (in a straight line) from O

To work out the distance back to the plane, we can map out the journey on a Coordinate Grid, like this:

Now it is just a matter of finding the distance between two points

The distance between the points (x_{A},y_{A})
and (x_{B},y_{B}) is given by the formula:

c = √(x_{A} − x_{B})^{2} + (y_{A} − y_{B})^{2}

and one of those points is always the origin, which is at (0,0), so when x_{B} and y_{B} are zero we get:

^{2}+ y

^{2}

Example, for the point **E (−2, 3)**, x = −2 and y = 3, and so:

^{2}) + 3

^{2}= √4+9 = √13

Hopefully that will help make your job easier.

## Direction?

There is one more thing Jade must know: to get back to the plane, what compass bearing to use?

This will be covered in Activity: A Walk in the Desert 2