#### You may also like

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

##### Age 14 to 16 Challenge Level:

A bead slides along a rail. The rail is initially vertical but then curves.

Explore this idea with the interactivity below . Then scroll down to see the main challenge.

Things to know :

• The graph is not plotting ordinary speed against time, it is plotting vertical speed or velocity against time. Experiment with the interactivity until you can predict the graph shape a particular curve will produce.

• This rail is also amazingly smooth - there's no friction or any other resistance you need to consider.

• At the end of a run click on the bead to put it back at the top.

• Also you can drag the bead up the vertical part of the rail to give it a faster entry onto the curve.

• Just experiment and you'll soon see all the things you can control.

Full Screen Version

If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.

Here are graphs for 4 different curves.

Can you shape the rail to produce each graph?

Explain how the features of the graph connect with the bead's behaviour on the rail. Once you've really "nailed it" why not send us your solution.

Can you, as an extra challenge, get a drop rate that's absolutely constant. The vertical velocity stays at one fixed value.

Send us a screen shot to prove that you did it, plus some text drawing attention to the qualities of the rail curve that made it possible.

Good luck!