The length and the breadth of a rectangular field are in the ratio 5:4 and its area is 3380 `m^(2)`. Find the cost of fencing it at the rate of Rs. 75 per m.

Answer

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

### Step 1: Set up the ratio
The length (L) and breadth (B) of the rectangular field are in the ratio of 5:4. We can express this as:
- Length = 5x
- Breadth = 4x

### Step 2: Use the area formula
The area (A) of a rectangle is given by the formula:
\[ A = \text{Length} \times \text{Breadth} \]
Substituting the expressions for length and breadth, we have:
\[ A = (5x) \times (4x) = 20x^2 \]

### Step 3: Set up the equation with the given area
We know the area of the field is 3380 m², so we can set up the equation:
\[ 20x^2 = 3380 \]

### Step 4: Solve for x²
To find \( x^2 \), divide both sides of the equation by 20:
\[ x^2 = \frac{3380}{20} \]
\[ x^2 = 169 \]

### Step 5: Solve for x
Now, take the square root of both sides to find x:
\[ x = \sqrt{169} \]
\[ x = 13 \]

### Step 6: Find the length and breadth
Now that we have the value of x, we can find the length and breadth:
- Length = \( 5x = 5 \times 13 = 65 \) meters
- Breadth = \( 4x = 4 \times 13 = 52 \) meters

### Step 7: Calculate the perimeter
The perimeter (P) of a rectangle is given by the formula:
\[ P = 2 \times (\text{Length} + \text{Breadth}) \]
Substituting the values we found:
\[ P = 2 \times (65 + 52) \]
\[ P = 2 \times 117 \]
\[ P = 234 \] meters

### Step 8: Calculate the cost of fencing
The cost of fencing is given at the rate of Rs. 75 per meter. Therefore, the total cost (C) can be calculated as:
\[ C = \text{Perimeter} \times \text{Cost per meter} \]
\[ C = 234 \times 75 \]
\[ C = 17550 \] Rs.

### Final Answer
The cost of fencing the rectangular field is Rs. 17550.

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