A man in a rowboat must get from point A on one bank to a point B on the opposite bank of a river. The distance BC = a. The width of the river AC = b. At what minimum speed \( v_b \) relative to the water should the boat travel to reach point B? The current velocity is \( v_r \)

\( (v_b \, cos \,\theta +v_r) \,t = a \) → (1)

\(v_b \: sin \,\theta \, t = b \) → (2)

\({{v_b \, cos \,\theta +v_r} \over {v_b\, sin \,\theta}} = {a \over b}\)

\(b\;(v_b \, cos \,\theta+v_r)=a\,{v_b \,sin \,\theta}\) → (3)

\(\text{For minimum of } v_b\) \({{dv_b} \over {d\theta} }= 0\)

\(b({dv_b\over{d\theta}}\, cos\theta-v_bsin\theta) = a(v_bcos \,\theta+{{dv_b}\over{d\theta}}\, sin \,\theta)\) → (4)

\(-bv_b \, sin \,\theta = a\,v_b\,cos \,\theta  \)     or     \(tan \,\theta =-{a \over b}\)

\( b({v_b}{{-b} \over \sqrt{a^2+b^2}}+v_r) = {{av_b} {a \over {\sqrt{a^2+b^2}}}} \)

\({b\,v_r}=v_b{a^2 \over \sqrt{a^2+b^2}} + v_b{b^2 \over \sqrt{a^2+b^2}} \)

\( v_b={{b\,v_r} \over \sqrt{a^2+b^2}} \)

$$ (v_b \, cos \, \beta +v_r\, cos \,\alpha)t = \sqrt{a^2+b^2}$$ $$ v_b \, sin \, \beta - v_r\, sin \,\alpha = 0 $$ $$ v_b = { {v_r\, sin \,\alpha} \over { sin \,\beta}}$$ \(v_b \) is minimum when \( \beta = 90° \) $$ v_b = {v_r\, sin \,\alpha} = {{b\,v_r} \over \sqrt{a^2+b^2}}$$

$$ v_b = v_r\, sin \,\alpha; $$ $$ v_b = {{b\,v_r} \over \sqrt{a^2+b^2}}$$