### 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:

A series of vessels of uniform cross section look like letters when viewed face-on. They are 1cm thick, and the corners of the vessels have either whole or half cm values for their coordinates.

Water is poured slowly into each vessel through one of the holes at the top at a rate of 1 cm$^3$ per minute.
Seven of these vessels take the same time to fill up. Which are they?
Which one takes the longest to fill?
Which would fill up first?

The height of the pool of water in one of the vessels is measured and a chart of the height against time is plotted, as follows:

Which vessel does this chart correspond to? Can you explain what each part of the chart corresponds to?

The very observant solver might have noticed that the chart given is actually slightly inaccurate: at certain points it is an over-measurement and at certain points it is an under-measurement. Can you see where and why?

Make charts of height against time for some of the other letters.