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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.
Age 16 to 18
Challenge Level
In designing experiments it is important to be sure that the data measured will be sufficient to reach a desired conclusion. In each of the following situations, will the information provided always, sometimes or never be sufficient to determine the required information about the reactants? Provide a clear analysis in each case.
- A gaseous alcohol is completely combusted with an excess of oxygen at $400$K. The change in volume of gasses is measured. Can I determine the formula of the alcohol?
- A gaseous hydrocarbon is completely combusted with an excess of oxygen at $400$K. The change in volume of gasses is measured. The system is then cooled to $273$K and the change in volume of gasses measured. Can I determine the molecular formula of the hydrocarbon?
- A quantity of gas is contained in a piston which is configured to exert a fixed pressure on the gas. The volume of gas is measured at two different temperatures. Can I determine the number of moles of the gas present?
- A gaseous hydrocarbon with known molecular formula is combusted with an insufficient supply of oxygen, resulting in a mixture of H$_2$O, CO and CO$_2$, with no oxygen present. The initial volume of hydrocarbon gas and the change in total volume of gas are measured. Can I determine the initial volume of oxygen?
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.