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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?

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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?

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Star Gazing

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

Similar Cylinders

Age 14 to 16 Short Challenge Level:
Using scale factors
The length scale factor between the smaller cylinder and the larger cylinder is 2, so the volume scale fator between the smaller cylinder and the larger cylinder is 2$^3$=8.

So the smaller cylinder will contain $\frac{1}{9}$ of the total volume, and the larger cylinder will contain $\frac{8}{9}$ of the total volume.

The volume of the cube is $9^3$ cm$^3$, so the volume of the smaller cylinder is $\frac{1}{9}\times9^3=81$ cm$^3$.


Using expressions for the volumes of the two cylinders
Suppose the smaller cylinder has height $h$ cm, so the larger cylinder has height $2h$ cm. Then the volume of the smaller cylinder is $\pi\times2^2\times h=4\pi h$ cm$^3$ and the volume of the larger cylinder is $\pi\times4^2\times2h=32\pi h$ cm$^3$.

The volumes of the cylinders will have to add up to the volume of the cube, so $$\begin{align}4\pi h+32\pi h=&9^3\\
\Rightarrow 36\pi h=&729\\
\Rightarrow h=&\frac{729}{36\pi}=\frac{81}{4\pi}\end{align}$$

Substituting into the expression for the volume of the smaller cylinder gives $4\pi h=4\pi\times\dfrac{81}{4\pi}=81$. So the volume of the smaller cylinder is $81$ cm$^3$.
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