A square field of area 31684 sq. metres is to be enclosed with wire placed at 1,2,3,4 metres above the ground. What length of the wire will be required, if the length required for each wire is 5% greater than the perimeter of the field?

Options

A. 2099 m
B. 2909 m
C. 2990.4 m
D. 2090 m

Answer

To solve the problem step-by-step, we need to follow these steps:

### Step 1: Calculate the side length of the square field.
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Given that the area is 31,684 sq. metres, we can find the side length:
\[ \text{side} = \sqrt{31,684} \]
Calculating this gives:
\[ \text{side} = 178 \text{ metres} \]

### Step 2: Calculate the perimeter of the square field.
The perimeter \( P \) of a square is given by the formula:
\[ P = 4 \times \text{side} \]
Substituting the side length we found:
\[ P = 4 \times 178 = 712 \text{ metres} \]

### Step 3: Calculate the length of wire required, considering the 5% increase.
The length of wire required is 5% greater than the perimeter:
\[ \text{Length of wire} = P + 0.05 \times P \]
This can be simplified to:
\[ \text{Length of wire} = P \times (1 + 0.05) = P \times 1.05 \]
Substituting the perimeter:
\[ \text{Length of wire} = 712 \times 1.05 = 746.6 \text{ metres} \]

### Step 4: Calculate the total length of wire for all heights.
Since the wire is placed at 1, 2, 3, and 4 metres above the ground, we need to multiply the length of wire for one height by the number of heights:
\[ \text{Total length of wire} = 746.6 \times 4 = 2986.4 \text{ metres} \]

### Final Answer:
The total length of wire required is approximately **2986.4 metres**.

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