### 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 2

##### Stage: 4 Challenge Level:

This problem follows on from the problem Maths Filler 1, although it can be attempted independently from this.

A vessel is constructed from a connected sequence of block-letters (assume that corners of the vessel lie either on grid vertices or half way between grid vertices). Water is poured slowly into the hole on the left (marked in blue) at a rate of 1cm$^3$ per minute.

A scale on the left of the vessel measures the height $H(t)$ of the pool of fluid which forms.

How long will it take to fill the vessel to reach each of the markers on the left hand side?

Use this to plot an approximate height-time chart.

Extensions: Think about the issues involved in drawing a completely accurate height-time chart.
How would the chart change if we were to fill through a different hole?