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?

Weekly Problem 4 - 2012

Stage: 4 Challenge Level:

Imagine the liquid became solidified.
Rotate the can until the face which contains XY is horizontal.
Now take the horizontal cross-section half way up the can which meets the plane face of the "liquid" in the line AB.
Observing that AB bisects the radius PQ, rotate the upper section of the "liquid" about AB through 180 degrees. The new position is shaded darkly and fits into the bottom right hand quarter (with some space to spare).
The actual fraction can be shown to be approximately 0.21. The answer is therefore less than a quarter.

This problem is taken from the UKMT Mathematical Challenges.

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