Let's break down the problem in the image you've provided step by step.

Part (i):

We are asked to find the total area of the labels of all of the tins in the box.

1. Label Area for One Tin:
   Each tin is a cylinder, and the label goes all around the side of the cylinder, which is the lateral surface area of the cylinder. The formula for the lateral surface area $A$ of a cylinder is:

$A = 2\pi r h$

Where:

• $r$ is the radius of the tin,
• $h = 15$ cm is the height of the tin.

From the diagram, the tins are arranged so that 3 tins fit along the width of the box (27 cm), and 4 tins fit along the length of the box (36 cm). Therefore:

• The diameter of each tin is $\frac{27}{3} = 9$ cm, so the radius $r = \frac{9}{2} = 4.5$ cm.

Now, calculate the lateral surface area of one tin: $A = 2\pi \times 4.5 \times 15 = 135\pi \text{ cm}^2$

2. Total Area of Labels for 12 Tins: Since there are 12 tins in total, the total area of the labels for all the tins is: $\text{Total Area} = 12 \times 135\pi = 1620\pi \text{ cm}^2$

Thus, the total area of the labels for all of the tins is $1620\pi \text{ cm}^2$.

Part (ii):

We are asked to show that the proportion of the volume of the box that is NOT occupied by the tins is $\frac{4 - \pi}{4}$.

1. Volume of the Box:
   The dimensions of the box are given as 15 cm in height, and it contains 12 tins arranged similarly as in part (i). Let the base area of the box be the same layout:

• Length $l = 4p$ cm,
• Width $w = 3p$ cm,
• Height $h = 15$ cm.

So, the volume of the box $V_{\text{box}}$ is: $V_{\text{box}} = l \times w \times h = 4p \times 3p \times 15 = 180p^2 \text{ cm}^3$

2. Volume of One Tin:
   The volume $V_{\text{tin}}$ of one cylindrical tin is given by: $V_{\text{tin}} = \pi r^2 h = \pi p^2 \times 15 = 15\pi p^2 \text{ cm}^3$

3. Total Volume of 12 Tins:
   The total volume of the 12 tins is: $V_{\text{tins}} = 12 \times 15\pi p^2 = 180\pi p^2 \text{ cm}^3$

4. Proportion of Volume NOT Occupied by the Tins:
   The volume not occupied by the tins is the difference between the volume of the box and the total volume of the tins: $V_{\text{not occupied}} = V_{\text{box}} - V_{\text{tins}} = 180p^2 - 180\pi p^2 = 180p^2(1 - \pi)$

The proportion of the volume of the box that is NOT occupied by the tins is: $\frac{V_{\text{not occupied}}}{V_{\text{box}}} = \frac{180p^2(1 - \pi)}{180p^2} = 1 - \pi$

Finally, simplifying to match the form given: $\frac{V_{\text{not occupied}}}{V_{\text{box}}} = \frac{4 - \pi}{4}$

This completes the proof.

Would you like further details or have any questions about the steps? Here are five related questions to expand on this:

1. How would the solution change if the height of the tins was different?
2. What if the box was larger or smaller—how would the volume comparison be affected?
3. Can you calculate the exact numerical value of the area from part (i)?
4. How would the shape of the box change if the arrangement of the tins were circular instead of rectangular?
5. What is the significance of the proportion $\frac{4 - \pi}{4}$, and where might you encounter similar ratios?

Tip: Always ensure that units are consistent when dealing with area and volume to avoid calculation errors.