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# Practice Paper 1 Question 18

You travel on Earth's surface south $$n$$ miles, then east $$n$$ miles, then north $$n$$ miles and find yourself back where you started, without visiting any point more than twice. What is the closest you could have been to the south pole when you started? Assume Earth is a sphere with radius $$R > n$$.

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## Warm-up Questions

1. What is the distance between the two points of intersection of two circles of radius 1 centred 1 unit apart.

2. If instead of circles we had hollow spheres, what is the length of their intersection path?

3. How can you determine the height of a building if you can measure only:
• the length of shadows on the ground and elevations angles.
• the length of shadows on the ground. Hint: you are allowed to measure shadows of multiple objects (even known objects).

## Hints

• Hint 1
Read the question carefully. In what ways is it possible to return to the starting point (remember you cannot visit a point more than twice)?
• Hint 2
You need to do a full circle somewhere.
• Hint 3
Besides a full circle, what other paths must you take, and how are they connected to the circle?
• Hint 4
How does each stage of the path you must take affect the longitude and the latitude?
• Hint 5
The full path is shown in the figure below. What is the distance of the starting point to the south pole?
• Hint 6
Try to sketch all necessary quantities in a 2D figure instead of 3D
• Hint 7
Consider the angle form the centre of the Earth to a point of lowest latitude. How are the other quantities linked to this angle?

## Solution

If you are allowed to visit any point at most twice then the solution is given by the following path:

You go down $$n$$ miles on the same longitude, then you take a round trip of $$n$$ miles laterally on the same latitude until you get back to same point, then you travel up $$n$$ miles on the same longitude. Let $$D$$ be the distance we seek and let $$r$$ be the radius of the circle of equal latitude (whose length we know equals $$n$$). We have:

\begin{align} 2{\pi}r &= n,\\ r&=R\sin(\theta),\\ D &= n + R\theta, \\ D &= n + R\arcsin{\textstyle\frac{n}{2{\pi}R}}. \end{align}

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