There is a road of equal width all around a circular garden. The outer and inner circumferences of the road are 328 m and 200 m respectively. The area of the road will be :

1. 5376 m²

2. 5375 m²

3. 5374 m²

4. 5373 m²

Outer circumference (C₁) = 328 m

Inner circumference (C₂) = 200 m

Circumference = 2π.radius

Let outre radius be R meters and inner radius be r meters.

Calculating outer circumference,

$\Rightarrow \text{Outer circumference} = 2 \times \dfrac{22}{7} \times R \\[1em] \Rightarrow 328 = 2 \times \dfrac{22}{7} \times R \\[1em] \Rightarrow R = \dfrac{328 \times 7}{44} = \dfrac{574}{11} \text{ m}$

Calculating inner circumference,

$\Rightarrow \text{Inner circumference} = 2 \times \dfrac{22}{7} \times r \\[1em] \Rightarrow 200 = 2 \times \dfrac{22}{7} \times r \\[1em] \Rightarrow r = \dfrac{200 \times 7}{44} \\[1em] \Rightarrow r = \dfrac{350}{11} \text{ m}.$

Area = π(R² - r²)

= π(R + r)(R - r)

$= \dfrac{22}{7} \times \Big(\dfrac{574}{11} + \dfrac{350}{11}\Big) \times \Big(\dfrac{574}{11} - \dfrac{350}{11}\Big) \\[1em] = \dfrac{22}{7} \times \dfrac{924}{11} \times \dfrac{224}{11} \\[1em] = 2 \times 84 \times 32 \\[1em] = 5376 \text{ m}^2.$

Hence, option 1 is the correct option.