Bricks are 20cm long and 10cm high. How high could an arch be built
without mortar on a flat horizontal surface, to overhang by 1
metre? How big an overhang is it possible to make like this?
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
Given the graph of a supply network and the maximum capacity for
flow in each section find the maximum flow across the network.
Suppose that the distance between the two wheels on a bike is $1$ unit (note that this is a modelling assumption: see the foot of this problem for more details). The bike moves forward and steers from the front. The rear wheel of the bike traces a curve $y = f(x)$ in the plane for some function $f(x)$.
Find an algebraic expression for the path travelled by the front wheel in terms of $x$ and $f(x)$.
Further numerical exploration
Use a spreadsheet to plot the path of the front and back wheels when the back wheel follows the paths:
Examine the form of these curves. Can you identify any common themes? Can you make any conjectures about the curves? Can you prove any of these conjectures?