The problem involves calculating the cost of painting a rectangular room. Let's solve it step by step.

Given Data:

• Room dimensions: Length $= 13 \, \text{ft}$, Breadth $= 12 \, \text{ft}$, Height $= 10 \, \text{ft}$.
• There are 2 windows of size $3 \, \text{ft} \times 4 \, \text{ft}$ and 1 door of size $3 \, \text{ft} \times 6 \, \text{ft}$.
• Cost of painting is Rs. 40 per square foot.

Part (a):

We are asked to find the cost of painting the four walls and the ceiling, excluding the areas of the door and windows.

Step 1: Total Area of the Four Walls

The total surface area of the four walls can be calculated as: $\text{Wall Area} = 2 \times (L \times H) + 2 \times (B \times H)$ where $L$ is the length, $B$ is the breadth, and $H$ is the height.

$\text{Wall Area} = 2 \times (13 \times 10) + 2 \times (12 \times 10) = 2 \times 130 + 2 \times 120 = 260 + 240 = 500 \, \text{sq ft}$

Step 2: Area of the Ceiling

The area of the ceiling is: $\text{Ceiling Area} = L \times B = 13 \times 12 = 156 \, \text{sq ft}$

Step 3: Total Area of Windows and Door

The area of the two windows and one door is: $\text{Window Area} = 2 \times (3 \times 4) = 2 \times 12 = 24 \, \text{sq ft}$ $\text{Door Area} = 3 \times 6 = 18 \, \text{sq ft}$ $\text{Total Area of Door and Windows} = 24 + 18 = 42 \, \text{sq ft}$

Step 4: Paintable Area

The paintable area is the total area of the walls and ceiling minus the area of the door and windows: $\text{Paintable Area} = (\text{Wall Area} + \text{Ceiling Area}) - \text{Total Area of Door and Windows}$ $\text{Paintable Area} = (500 + 156) - 42 = 656 - 42 = 614 \, \text{sq ft}$

Step 5: Cost of Painting

The cost of painting the room is: $\text{Cost} = \text{Paintable Area} \times \text{Cost per sq ft}$ $\text{Cost} = 614 \times 40 = 24,560 \, \text{Rs.}$

Thus, the cost of painting the walls and ceiling is Rs. 24,560.

Part (b):

We are asked to find the increase in cost if the rate per square foot is increased by 25%.

Step 6: New Cost per Square Foot

If the cost per square foot is increased by 25%, the new cost per square foot will be: $\text{New Cost per sq ft} = 40 + 0.25 \times 40 = 40 + 10 = 50 \, \text{Rs.}$

Step 7: New Total Cost

The new total cost will be: $\text{New Cost} = 614 \times 50 = 30,700 \, \text{Rs.}$

Step 8: Increase in Cost

The increase in cost is: $\text{Increase} = \text{New Cost} - \text{Old Cost} = 30,700 - 24,560 = 6,140 \, \text{Rs.}$

Thus, the total cost will increase by Rs. 6,140 if the rate increases by 25%.

Would you like more details on any part of this solution?

Related Questions:

1. How is the total surface area of a cuboid calculated?
2. How would the cost change if the room had additional windows or doors?
3. What is the formula for calculating the area of a rectangular surface?
4. How does a percentage increase in cost affect the total expense?
5. What would the total cost be if the rate increased by 50% instead of 25%?

Tip: When calculating costs involving increases or decreases, always apply the percentage change directly to the original rate to avoid confusion.