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### Mediant

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

# Perfect Eclipse

##### Stage: 4 Challenge Level:

A total solar eclipse is an amazing spectacle with the sun appearing to move exactly behind the moon for a few moments, with the view of the moon seeming to perfectly overlap the sun.

But how perfect is this coverage? Is the view of the moon slightly larger than that of the sun or vice versa? In other words, exactly how perfect is a total solar eclipse as viewed from earth?

To answer this question you will need to work out the apparent size or angular diameter of the sun and the moon as viewed from Earth, using the following astronomical data:

Radius of the sun: $695,500$ km

Radius of the moon: $1,737$ km

Radius of the earth: $6,371$ km

Smallest distance between Earth and Sun $147,098,074$ km

Largest distance between Earth and Sun $152,097,701$ km

Smallest distance between Moon and Earth $356,375$ km

Largest distance between Moon and Earth $406,720$ km

Here are some issues to consider:

• How much does the apparent size of the moon change, as its distance from the Earth changes?
• How much does the apparent size of the sun change?
• Can there ever be a perfect eclipse?

Extension:

Could other planets have perfect eclipses? To help you address this question you can consider the following data for the moons of other planets in the solar system.

NOTES AND BACKGROUND

A solar eclipse seen from Earth is one of our solar system's most beautiful sights. From the moment the Moon begins to cross into the Sun's visible disc, the light covers the Moon in a special way, such that you have a feeling of the Moon's depth; the Moon as a near-sphere, rather than the disc we usually perceive. The process of the Moon slowly covering the Sun then takes a long time, as you have a perception of this astronomical event. Then as the Moon just begins to cover the Sun there is an amazing "diamond ring" effect for a short time as the final rays of light are visible. Then for a few minutes the situation seems to remain still, and only at this time you can see a shimmering corona of particles that are around the sun, patterned by powerful forces, at millions of degrees Kelvin. Then finally the diamond ring re-emerges, and the whole spectacle is seen in reverse. If you are lucky you might see one in your lifetime. Some people travel to wherever in the world one will be visible. Often people speak of the "eclipse coincidence", the fact that it is actually rather unlikely to have this spectacle on Earth, which we already believe to be quite a "lucky" planet, having just the right raw materials and being just the right distance from a star of just the right age to support our life.