Question

A river is flowing with a speed of $1\text{km/hr}$. A swimmer wants to go to point $C$ starting from $A$. He swims with a speed of $5\text{km/hr}$ with respect to river flow at angle $\theta$ as shown. If $AB=BC=400\text{m}$, the value of $\theta$ is

• A. ${37}^{\circ }$
• B. ${30}^{\circ }$
• C. ${53}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

The correct option is C ${53}^{\circ }$

Time to cross river $\left(t\right)=\frac{AB}{v_{mr}\sin \theta }=\frac{0.4}{5\sin \theta }$

Drift
$BC=(v_{mr}\cos \theta +v_r)t$
$\Rightarrow 0.4=(5\cos \theta +1)\times \frac{0.4}{5\sin \theta }\Rightarrow 5\sin \theta -5\cos \theta =1$
$\Rightarrow 25{\sin }^2\theta +25{\cos }^2\theta -50\sin \theta \cos \theta =1\Rightarrow 25\sin 2\theta =24$

$\Rightarrow \sin 2\theta =\frac{24}{25}$

$\Rightarrow \theta ={53}^{\circ }$

Hence, option $\left(\text{c}\right)$ is correct.