A boat on one bank of the river wants to reach other bank of river flowing with speed v0. Speed of boat in still water is also v0. Boat moves in such a way that its distance from a fixed point P remains constant equal to 2d . Time taken to cross the river is dv0ln⁡x+x2−1, value of x, is

dt=2ddθ2v0cos⁡θt=dv0∫0π/3 sec⁡θdθ=dv0ln⁡|(sec⁡θ+tan⁡θ)|0π/3=dv0ln⁡(3+2)−ln⁡(1)=dv0ln⁡(2+3)

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A boat on one bank of the river wants to reach other bank of river flowing with speed v₀. Speed of boat in still water is also v₀. Boat moves in such a way that its distance from a fixed point P remains constant equal to 2d . Time taken to cross the river is $\frac{d}{v_{0}} \ln (x + \sqrt{x^{2} - 1})$ , value of x, is

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Detailed Solution

$\begin{array}{l} dt = \frac{2 d (dθ)}{2 v_{0} \cos θ} t = \frac{d}{v_{0}} \int_{0}^{π / 3} \sec θdθ \\ = \frac{d}{v_{0}} \ln | (\sec θ + \tan θ) |_{0}^{π / 3} = \frac{d}{v_{0}} \ln (\sqrt{3} + 2) - \ln (1) = \frac{d}{v_{0}} \ln (2 + \sqrt{3}) \end{array}$