Question

Deserted on a deserted Island you spot a slightly exposed tin can under a tree. Upon opening it you find it’s instructions to a treasure. It reads: “Ten paces from this very tree in a direction twenty degrees south of west lies the first location. Ten paces from this very tree in a direction sixty degrees north of east lie the second location. Walk from this tree exactly the distance and direction you would walk from the first location to the second location and you will find ye treasure. Yar” (a) What are the coordinates of treasure? (b) Use a physical representation in conjunction with the Related Quantities sense-making technique to check the validity of your answer.

Answer 1

A pace is a unit of length consisting either of one normal walking step (~0.75 metres or 0.82 yards), or of a double step, returning to the same foot (~1.5 metres or 1.6 yards). In the US, it is an uncommon customary unit of length denoting a brisk single step and equal to 2 ¹⁄₂ feet or 30.0 inches. Here, I will consider the length unit as paces (without converting it to ft ot others) and I will calculate the coordinates of the points in the same unit system.

Let us consider, the position of the tree is at origin O. The location of first and second position of treasure are A and B respectively.

The figure shows the two points of the location of treasure.

(a) Let us consider the coordinates of the points A and B are (-x₁, -y₁) and (x₂, y₂)

Now, we can calculate the coordinates from trigonometric identities,

x₁ = 10 cos 20 = 9.39 paces

y₁ = 10 sin 20 = 3.42 paces

x₂ = 10 cos 60 = 5.00 paces

y₂ = 10 sin 60 = 8.66 paces

Therefore, the coordinates of first treasure is A (-9.39, -3.42) and the coordinates of second treasure is B (5.00, 8.66).

(b) The distance between two treasure point is

Then, we can say that you have to travel from origin to first treasure and then you have to move directly towards the second treasure. Hence, you should travel a total distance of S_OA + S_AB = 10 + 18.79 = 28.79 paces

Direction wise, you have to go 10 paces from this very tree in a direction twenty degrees south of west and then 18.79 paces towards forty degree north of east.