### Dividing the Field

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

### Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# Candy Floss

##### Age 14 to 16 Short Challenge Level:
The volume of a cylinder is given by cross-sectional area $\times$ length, so $\pi r^2\times \text{length}$.
Working in mm, the original cylinder of sugar has volume $\pi \left(\dfrac{30}{2}\right)^2\times 40mm^3$

Finding the volume of sugar as a number
$\pi \left(\dfrac{30}{2}\right)^2\times 40=\pi\dfrac{30^2}{4}\times40=\pi\times900\times10=9000\pi$.
The cylinder of candy floss has volume $\pi \left(\dfrac{1}{2}\right)^2 x$, so \begin{align}\pi \left(\frac{1}{2}\right)^2 x=&9000\pi\\ \Rightarrow\pi\times\frac{1}{4}x=&9000\pi\\ \Rightarrow x=&9000\times4\\ \Rightarrow x=&36000\end{align}So $x=36000$mm, or $36$m.

Using algebra to find the length directly
The cylinder of candy floss has volume $\pi \left(\dfrac{1}{2}\right)^2 x$, so \begin{align}\pi \left(\frac{1}{2}\right)^2 x&=\pi \left(\frac{30}{2}\right)^2\times 40\\ \Rightarrow 1^2x&=30^2\times40\\ \Rightarrow x&=36000\end{align}
So $x=36000$mm, or $36$m.

Using scale factors of enlargement
The ratio of the diameters of the cylinders is 1:30,
so the ratio of the cross-sectional areas is 1:900,
so the height of the thin cylinder will need to 900 times longer than the height of the fat cylinder.

Therefore the height of the thin cylinder is 36m long.

You can find more short problems, arranged by curriculum topic, in our short problems collection.