### 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:

The first 6 marks are filled one every half minute.

Then for mark 7 we first fill up to 6 completely:

$M+ = 1+3+1 = 5$
$A+ = \frac{3.5+4}{2} \times 2 +4 - \frac{1+2}{2} \times 2 +2 = 10.5$
$T+ = 6+2 = 8$
$H+ = 3+3+3+1 = 10$
$S = 4+1+1+1+1+0.5 = 8.5$
42 total

And then fill up to 7:

$M = 2$
$A = \frac{3.5 + 3.375}{2} \times 0.5 -1 \times 0.5 = \frac{39}{32}$
$T = 2$
$H = 2$
$S = 2 \times \frac{1}{2} + \frac{1}{8} = \frac{9}{8}$

$\frac{267}{32}$ total (sum to this point = $\frac{1707}{32}$)

And following the same process for the rest of the levels:

Level 8: $\frac{225}{32}$ total (sum to this point = $\frac{1932}{32}$)

level 9: $\frac{267}{32}$ total (sum to this point = $\frac{2199}{32}$)

Level 10: $\frac{249}{32}$ total (sum to this point = 76.5)

The sum of all these is 76.5.

Now if we check the total size to check it's the same: $24 \times 5 - 23.5 \textrm{ spaces} = 76.5$ check!

So we can plot these points on a graph! Joining them up is another matter...