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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%.

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### 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.

The length, breadth, and height of a rectangular room are 13 ft, 12 ft, and 10 ft, respectively. There are 2 windows of size 3 ft × 4 ft and a door of size 3 ft × 6 ft. How much does it cost to paint the four walls and ceiling excluding the door and windows at the rate of Rs. 40 per square foot? By how much will the total cost of painting increase if the rate is increased by 25%?

Calculate the cost of painting the walls and ceiling of a room with dimensions 13 ft × 12 ft × 10 ft, excluding the areas of two windows and a door. Learn how to account for a 25% price increase per square foot. Perfect for students in Grades 8-10.

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