### 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?

# Maths Filler

##### Stage: 4 Challenge Level:
Well done to Anurag and Christina for their solutions to this problem:

The seven letters that take the same time to fill up are: I,L,O,E,M,T and H, all with a volume $14$cm$^3$ and thus taking $14$ minutes to fill.

The letter S takes the longest to fill up ($14.5$cm$^3$, $14.5$ minutes to fill).
The letter V fills up first ($13$cm$^3$, $13$ minutes to fill).
The letter A will take $13.5$ minutes to fill.

The graph corresponds to the letter M. Some points in the graph are over measured, for instance, the points between 4 and 5 cm.

1. 0 - 3 minutes - filling one 'leg' of M, with rate of height increase constant due to constant width

2. 3 - 7 minutes - further water will run over in to the central dip of the M, and then once this is filled into the opposite leg. These have a combined volume of $4$cm$^3$ and so take 4 minutes to fill

3. 7 - 11 minutes - water fills top rectangular section with constant rate of height increase

4. 11 - 14 minutes - filling up top two trapezoidal sections of M

5. 14 - 16 minutes - letter completely full - no further height gain

Section 4 in the period 11-14 minutes should actually be represented by a curved line on the chart as the width of the section being filled is changing with height, and therefore so will the rate of height increase.