Gillian wants to make cylindrical soy wax candles as farewell gifts for 50 classmates and friends in school. The diagram below shows one of the cylindrical candles she intends to make. Gillian's mum has the tools for candle-making and is willing to provide 50 cylindrical jars for her farewell gifts. The following tables give online information that Gillian can use to buy the rest of the materials that she needs for making the 50 soy wax candles.

1. Find the wick length needed for one candle and divide the available length by it.

Length of wick for one candle
$=8\text{ cm}+12\text{ mm}$
Since $12\text{ mm}=1.2\text{ cm}$,
$\text{length per candle}=8+1.2=9.2\text{ cm}.$
Also,
$10\text{ m}=1000\text{ cm}.$
Number of candles
$=\frac{1000}{9.2}=108.69\ldots$
Hence, the number of complete candles that can be made is
$108.$

1. Convert the wick diameter to a radius in centimetres.

Diameter of wick $=4\text{ mm}=0.4\text{ cm}$

Radius of wick $=\dfrac{0.4}{2}=0.2\text{ cm}$

2. Find the volume of the cylindrical wick from A to B.

Height from A to B $=8\text{ cm}$

$\text{Volume}=\pi r^2h$

$=\pi(0.2)^2(8)$

$=0.32\pi$

$=1.005\ldots\text{ cm}^3$

1. Calculate the internal volume of the cylindrical jar up to the candle height.

The jar is a cylinder with radius $2.9\text{ cm}$ and height $8\text{ cm}$.

$\text{Volume of jar}=\pi r^2h$

$=\pi(2.9)^2(8)$

$=211.366\ldots\text{ cm}^3$

2. Subtract the volume of the wick inside the candle to get the volume of soy wax needed.

From part (b), volume of wick inside the candle $=0.1005\text{ cm}^3$.

$\text{Volume of soy wax}=211.366\ldots-0.1005$

$=211.265\ldots\text{ cm}^3$

$\approx 211\text{ cm}^3$

1. Find the total mass of soy wax needed for 50 candles.

From part (c), volume of soy wax for one candle $\approx 211\text{ cm}^3$.

For 50 candles,
$\text{total volume}=50\times 211=10550\text{ cm}^3.$

Density of soy wax $=0.9\text{ g/cm}^3$,
$\text{mass}=10550\times 0.9=9495\text{ g}=9.495\text{ kg}.$

2. Decide the soy wax boxes to buy and calculate their cost.

Gillian needs at least $9.495\text{ kg}$ of soy wax.

The cheapest sufficient purchase is $2$ boxes of $5\text{ kg}$ soy wax:
$2\times \$64.73=\$129.46.$

Since $\char36{}129.46>\char36{}45$, shipping for the soy wax is free.

3. Find the cost of the cotton wick needed.

From part (a), a $10\text{ m}$ length of cotton wick is enough for $50$ candles.

So Gillian needs to buy one $10\text{ m}$ length of cotton wick.

$\text{cost of wick}=\$6.32+\$1=\$7.32.$

4. Find the cost of the wick tabs and stickers needed.

She needs $50$ wick tabs and stickers, but the fixed minimum purchase is $100$.

$\text{cost of tabs and stickers}=\$1.20+\$1=\$2.20.$

5. Calculate total cost and cost per gift, then judge the claim.

Total cost:
$\$129.46+\$7.32+\$2.20=\$138.98.$

Cost per farewell gift:
$\frac{\$138.98}{50}=\$2.7796\approx \$2.78.$

Since $\char36{}2.78<\char36{}3$, each farewell gift will not cost Gillian more than $\char36{}3$.