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Qqq..cubed

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

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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?

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In a Spin

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

Maths Filler

Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem requires careful thought about the way the water level in a vessel changes when water is added at a constant rate. Through analysing the key features of a graph, students can figure out the shape of the vessel it represents.

Possible approach

The first part involves working out volumes. The key is to realise that the cross-sectional area is proportional to the volume and then to work out the areas. There are obvious 'easy' candidates for this and some harder letters. There are various ways in which the areas of the cross sections of the vessels can be 'rearranged' to form rectangles. Students could work on finding the areas in small groups and then feed back to the rest of the class, sharing their approaches for finding the trickier areas.

To work out which letter the graph corresponds to, ask for suggestions for a 'story' relating the height-chart diagram to a vessel filling up. For example, what happens to the water level at the horizontal parts of the graph? What could be happening to account for this?

Once the class have identified the correct vessel for the graph, they could work on producing graphs for the other letters. Students could check each other's work by seeing if they can match the graphs with the vessels.

Small groups of students could also design some other letters in the same way and draw the resulting graphs, perhaps producing a card-matching activity to challenge other groups. The results could contribute to a classroom display.

Key questions

What could be happening at the horizontal parts of the height-time graph?
What can you work out from the steepness of the lines on the graph?

Possible extension

Would the graphs change if the holes were moved, or if water was poured into both holes where available?

The final part of the M graph should be a curve rather than a straight line. Can students justify why the graphs for V, A, and S will also contain curves?

Can students work out the functions which describe any of these curved parts?

See also Maths Filler 2 for a suitable follow-up challenge.

Possible support

How Far Does it Move? might provide a useful introduction to interpretation of graphs.

Start by working on the letters without diagonal lines and work out how quickly they will fill up.