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Three Cubes

Can you work out the dimensions of the three cubes?

Concrete Calculation

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?


Age 14 to 16
Challenge Level

See the Hint section for detailed support for students, but one of the main aims of this problem is to use a spreadsheet to open up a problem that is otherwise rather tedious to pursue by calculation. Students may need help thinking through the calculation required and the spreadsheet commands (formulae) necessary to achieve that. This is a challenging problem.

The relationship between surface area and volume enclosed is an important theme for able students at Stage 4 and using IT effectively should be part of a good student's repertoire of problem solving approaches.

With able students it is good to draw out that the ratio between the volume and the surface area will change with scale, but the optimum shape for the cone will not.
Perhaps reasoning in terms of units : measurements in a different unit of length would change all the numbers (how ?) but not the height-radius ratio (or cone angle) where the surface area takes its minimum value.