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Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Diamonds Aren't Forever

Age 16 to 18
Challenge Level

Remember that the standard units for use in the ideal gas equation are as follows:

Pressure is in Pascals (Pa) or Nm$^{-2}$
Volume is in m$^3$
T is in Kelvins (K), which is calculated by K = 273 +$^\circ$C

If the gas is not expanded against a vacuum, think about whether it will be harder for the gas to expand or not? Consequently, think about how the energy of the gas will change, and therefore what will happen to the temperature and thus pressure.

For the final part of the problem, think about using the given radii for the two atoms. Either model each atom/molecule as a sphere or cuboid, and thus estimate its volume. Assume that the point of minimum volume is where there is no space between the atoms. Therefore, multiply the volume of a single particle by the total number of particles.