The area of a square field is 8 hectare. How long would a man take to cross it diagonally by walking at the rate of 4 kmph?

Given,

Area of square field = 8 hectares

Speed of man = 4 kmph

We know,

1 hectare = 10,000 m²

8 hectares = 8 × 10,000 = 80,000 m²

$\Rightarrow \text{Area of a square} = \text{ (side)}^2 \\[1em] \Rightarrow \text{(side)}^2 = 80,000 \\[1em] \Rightarrow \text{ side} = \sqrt{80000} \\[1em] \Rightarrow \text{ side} = \sqrt{8 × 10000} \\[1em] \Rightarrow \text{ side} = \sqrt{8} \times \sqrt{10000} \\[1em] \Rightarrow \text{ side} = 2 \sqrt{2} \times 100 \\[1em] \Rightarrow \text{ side} = 200 \sqrt{2} \text{ m}.$

Diagonal of a square = side × $\sqrt{2}$

= 200 $\sqrt{2} \times \sqrt{2}$

= 200 × 2 = 400 m = $\dfrac{400}{1000}$ = 0.4 km

We know that,

Time = $\dfrac{\text{Distance}}{\text{Speed}} = \dfrac{0.4}{4} = \dfrac{1}{10}$ hr.

1 hr = 60 min

∴ $\dfrac{1}{10} \text{ hr} = \dfrac{1}{10}$ × 60 min = 6 minutes.

Hence, a man takes 6 min to cross it diagonally.