Problem Analysis:

1. Shape of the room: The room is a cuboid with dimensions:

   • Length: $2.8 \, \text{m}$
   • Width: $2.0 \, \text{m}$
   • Height: $2.2 \, \text{m}$

2. Areas to be painted:

   • The 4 walls and the ceiling.

3. Paint details:

   • Cost of a tin of paint: £49.99
   • Coverage per tin: $3 \, \text{m}^2$

Step-by-Step Solution:

Step 1: Calculate the surface area of the walls

The room has four walls, which include:

• Two walls of dimensions $2.8 \, \text{m} \times 2.2 \, \text{m}$
• Two walls of dimensions $2.0 \, \text{m} \times 2.2 \, \text{m}$

The total area of the walls is: $2 \times (2.8 \times 2.2) + 2 \times (2.0 \times 2.2)$

$= 2 \times 6.16 + 2 \times 4.4 = 12.32 + 8.8 = 21.12 \, \text{m}^2$

Step 2: Calculate the surface area of the ceiling

The ceiling has dimensions $2.8 \, \text{m} \times 2.0 \, \text{m}$. Its area is: $2.8 \times 2.0 = 5.6 \, \text{m}^2$

Step 3: Total surface area to paint

The total area is: $21.12 + 5.6 = 26.72 \, \text{m}^2$

Step 4: Calculate the number of tins of paint needed

Each tin covers $3 \, \text{m}^2$. The number of tins required is: $\lceil \frac{26.72}{3} \rceil = \lceil 8.9067 \rceil = 9 \, \text{tins}$

Step 5: Calculate the total cost

The cost of one tin is £49.99. The total cost is: $9 \times 49.99 = 449.91 \, \text{GBP}$

Final Answer:

The total cost of the paint needed to paint the 4 walls and the ceiling is £449.91.

Do you need further explanation or details?

Related Questions:

1. How would the cost change if each tin covered $4 \, \text{m}^2$?
2. What would be the cost for painting only the walls, excluding the ceiling?
3. If the room height was reduced to $2.0 \, \text{m}$, how many tins would be required?
4. How can you minimize leftover paint in this situation?
5. Can you calculate the cost of painting two such rooms?

Tip:

Always round up when calculating the number of tins needed, as partial tins cannot be purchased.