Find the equation of the intersection points (sketched in red) of the straight lines drawn as shown.

In the diagram shown: If a straight line crosses the x axis at 1, it will cross the y axis at 7. If it crosses the x axis at 2, it will cross the y axis at 6. For a general real number between 0 and 8, if a straight line crosses the x axis at point x, it will cross the y axis at 8-x.

This graph as shown is only valid on the x and y range 0 to 8. You will have to generalize this so that the graph is valid between the range x=0 and x=a. Consider the curve formed by infinitely many intersection points, of a curve valid on the entire range of the x and y axes. Find the equation of this curve (sketched in red).

How does the melting of the polar ice caps affect the length of a day on Earth?

A boat floats ontop of a water bridge (not a bridge over water, a bridge for water to flow over for boats to cross) connecting two very large bodies of water. The bridge can support a mass of 50,000 kg. The water flowing over the bridge has mass 47,000 kg.

A boat of mass 4,000 kg begins to cross the bridge.

What will happen to the bridge when the boat crosses?

If there are ‘n’ people in a room, and each person wants to shake hands with every other person in the room. How many handshakes in total take place?

How about, if we imagine some bizarre handshake, taking part between multiple people simultaneously (we will call it a k-shake. Where each shake requires ‘k’ people to take part). How many possible k shakes are there in a room of ‘n’ people?

When I drop a pencil from around waist height, how much does the Earth move in reponse?

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