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Let’s assume we have to run at an angle ‘θ’ w.r.t. the position of the dog and the centre of the circle. Therefore, Man’s velocity along radius = v.cosθ Displacement along the radius = R 𝑅 Time taken = 𝑣.𝑐𝑜𝑠θ (1) Angular velocity of man = v.sinθ To not get caught by the dog, (v.sinθ).t + π.R > 4.v.t From (1), R = v.cosθ.t Therefore, v.sinθ.t + π.v.t.cosθ > 4vt sinθ + π.cosθ > 4 Dividing both sides by 1+π², 1 sinθ 1+π² + 𝜋 cosθ 1+π² > 4 1+π² 1 Sinα = 1+π², Sinαsinθ + cosαcosθ = cos(α-θ) > 4 α-θ < 𝑐𝑜𝑠 −1 1+π² 4 ...
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