A piece of wood which was 50 cm long, was in the shape of a uniform cylinder. A prism is cut out from the wood such that its cross-section is a minor segment of a circle, with centre O, radius r cm and $\angle AOB = \frac{7}{9}\pi$ radians.

1. Convert the given area into square centimetres

Since $1\text{ m}=100\text{ cm}$,
$1\text{ m}^2=10000\text{ cm}^2.$
Given area
$=3\times10^{-3}\text{ m}^2=3\times10^{-3}\times10000=30\text{ cm}^2.$

2. Form the area expression for the minor segment

Angle of sector:
$\theta=\frac{7\pi}{9}.$
Area of minor segment
$=\text{area of sector }AOB-\text{area of }\triangle AOB$
$=\frac12 r^2\theta-\frac12 r^2\sin\theta.$
So
$30=\frac12 r^2\left(\frac{7\pi}{9}-\sin\frac{7\pi}{9}\right).$

3. Solve for r and round to 3 decimal places

$r^2=\frac{60}{\frac{7\pi}{9}-\sin\frac{7\pi}{9}}$
$r=\sqrt{\frac{60}{\frac{7\pi}{9}-\sin\frac{7\pi}{9}}}$
$r=5.772\ldots$
Therefore,
$r=5.772\text{ cm} \quad \text{correct to 3 decimal places}.$

1. Identify the six edges to be covered by ribbon

The cross-section of the prism is a minor segment bounded by chord $AB$ and the minor arc $AB$.

The six edges are:

• two length edges of $50$ cm,
• two chord edges $AB$,
• two arc edges $AB$.

2. Find the chord length $AB$

Using $r=5.772$ cm and $\angle AOB=\dfrac{7\pi}{9}$,

$AB=2r\sin\left(\frac{1}{2}\times \frac{7\pi}{9}\right)$

$AB=2(5.772)\sin\left(\frac{7\pi}{18}\right)$

$AB=10.846\text{ cm} \;\text{(approximately)}$

3. Find the arc length $AB$

Arc length $=r\theta$.

$\text{arc }AB=5.772\times \frac{7\pi}{9}$

$\text{arc }AB=14.104\text{ cm} \;\text{(approximately)}$

4. Find the total ribbon needed and compare with 2 m

Total length of ribbon needed

$=2(50)+2(10.846)+2(14.104)$

$=149.9\text{ cm}$

Since $2$ m $=200$ cm,

$149.9<200.$

Therefore, there is sufficient ribbon.