### 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?

### 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?

# Immersion

##### Stage: 4 Challenge Level:

Consider these solids:

1) A sphere of radius 1cm
2) A solid cylinder with height $\frac{4}{3}$cm and radius $1$cm
3) A solid circular cone with base radius $1$cm and height $4$cm.
4) A solid cylinder of height $\frac{4}{9}$cm with a hole drilled through it, leaving an annular (ring-shaped) cross-section with internal radius $1$cm and external radius $2$cm.

Can you sketch what each solid would look like?
Can you work out the volume of each solid?

Experiments are conducted where a solid is chosen and has a string firmly attached at a fixed point. The solid is then lowered at a rate of 1cm per minute into a beaker of water and the height of displaced water measured, with graphs of height against time drawn.

The results are measured on the following chart:

Can you work out what the two axes represent?

Can you work out which curve corresponds to which solid and in which orientation it is lowered into the beaker? (Note: One solid is used twice, in two different orientations).

Could you sketch the curve for the same solids in other orientations? What about different solids?

Extension task: Can you find equations which represent the volumes of the immersed parts of the solids? They vary in difficulty; if you cannot find the equation explicity, can you describe clearly what needs to be found? Reproduce as much of the above graph as you can.