A display board is in the shape of a regular octagon. If the cost of putting a border around its edges at the rate of ₹ 3 per metre is ₹ 96, find the length of one side of the board.

Answer

To solve the problem step by step, we will follow the information given in the question and apply the relevant formulas.

### Step 1: Understand the relationship between cost, perimeter, and cost per meter.
We know that the cost of putting a border around the octagon is given by the formula:
\[ \text{Cost} = \text{Perimeter} \times \text{Cost per meter} \]
From the question, we have:
- Cost = ₹ 96
- Cost per meter = ₹ 3

### Step 2: Set up the equation.
Using the formula from Step 1, we can set up the equation:
\[ 96 = \text{Perimeter} \times 3 \]

### Step 3: Solve for the perimeter.
To find the perimeter, we can rearrange the equation:
\[ \text{Perimeter} = \frac{96}{3} \]
Calculating this gives:
\[ \text{Perimeter} = 32 \text{ meters} \]

### Step 4: Use the formula for the perimeter of a regular octagon.
The perimeter \( P \) of a regular octagon can be calculated using the formula:
\[ P = 8 \times \text{length of one side} \]
Let the length of one side be \( s \). Therefore, we can write:
\[ 32 = 8s \]

### Step 5: Solve for the length of one side.
Now, we can solve for \( s \):
\[ s = \frac{32}{8} \]
Calculating this gives:
\[ s = 4 \text{ meters} \]

### Conclusion:
The length of one side of the board is **4 meters**.

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