A school has a hall which is 22 m long and 15.5 m broad. A carpet is laid inside the hall leaving all around a margin of 75 cm from the walls. Find the area of the carpet and the area of the strip left uncovered. If the width of the carpet is 82 cm, find its cost at the rate of rs.60 per m.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Calculate the area of the hall.
The area of a rectangle is calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]

Given:
- Length of the hall = 22 m
- Breadth of the hall = 15.5 m

Calculating the area:
\[ \text{Area of the hall} = 22 \, \text{m} \times 15.5 \, \text{m} = 341 \, \text{m}^2 \]

### Step 2: Convert the margin from centimeters to meters.
The margin left around the carpet is given as 75 cm. We need to convert this to meters:
\[ 75 \, \text{cm} = \frac{75}{100} \, \text{m} = 0.75 \, \text{m} \]

### Step 3: Calculate the dimensions of the carpet.
To find the dimensions of the carpet, we need to subtract the margin from both sides of the length and breadth of the hall.

**Length of the carpet:**
\[ \text{Length of the carpet} = \text{Length of the hall} - 2 \times \text{Margin} \]
\[ = 22 \, \text{m} - 2 \times 0.75 \, \text{m} \]
\[ = 22 \, \text{m} - 1.5 \, \text{m} = 20.5 \, \text{m} \]

**Breadth of the carpet:**
\[ \text{Breadth of the carpet} = \text{Breadth of the hall} - 2 \times \text{Margin} \]
\[ = 15.5 \, \text{m} - 2 \times 0.75 \, \text{m} \]
\[ = 15.5 \, \text{m} - 1.5 \, \text{m} = 14 \, \text{m} \]

### Step 4: Calculate the area of the carpet.
Now that we have the dimensions of the carpet, we can calculate its area:
\[ \text{Area of the carpet} = \text{Length of the carpet} \times \text{Breadth of the carpet} \]
\[ = 20.5 \, \text{m} \times 14 \, \text{m} = 287 \, \text{m}^2 \]

### Step 5: Calculate the area of the uncovered strip.
To find the area of the uncovered strip, we subtract the area of the carpet from the area of the hall:
\[ \text{Area of the uncovered strip} = \text{Area of the hall} - \text{Area of the carpet} \]
\[ = 341 \, \text{m}^2 - 287 \, \text{m}^2 = 54 \, \text{m}^2 \]

### Step 6: Calculate the cost of the carpet.
The width of the carpet is given as 82 cm, which needs to be converted to meters:
\[ 82 \, \text{cm} = \frac{82}{100} \, \text{m} = 0.82 \, \text{m} \]

Now, we can find the length of the carpet in meters using the area:
\[ \text{Length} = \frac{\text{Area of the carpet}}{\text{Width}} \]
\[ = \frac{287 \, \text{m}^2}{0.82 \, \text{m}} \approx 350 \, \text{m} \]

Finally, we calculate the cost of the carpet at the rate of Rs. 60 per meter:
\[ \text{Total cost} = \text{Length} \times \text{Cost per meter} \]
\[ = 350 \, \text{m} \times 60 \, \text{Rs/m} = 21000 \, \text{Rs} \]

### Summary of Results:
- Area of the carpet: **287 m²**
- Area of the uncovered strip: **54 m²**
- Total cost of the carpet: **21000 Rs**