# Candy floss

What length of candy floss can Rita spin from her cylinder of sugar?

## Problem

Rita is making candy floss at home.

She melts a cylinder of sugar with diameter 3 cm and length 4 cm.

She then spins it into a long cylindrical strand of diameter 1 mm.

Assuming that the melting and spinning does not affect the volume of the sugar, how long is the strand of candy floss?

## Student Solutions

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 $\pi \left(\dfrac{30}{2}\right)^2\times 40mm^3$

$\pi \left(\dfrac{30}{2}\right)^2\times 40=\pi\dfrac{30^2}{4}\times40=\pi\times900\times10=9000\pi$.

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.

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.

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.

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.