21. Моторная лодка прошла против течения реки 117 км и вернулась в пункт отправления, затратив на обратный путь на 2 часа меньше, чем путь против течения. Найдите скорость лодки в неподвижной воде, если скорость течения реки равна 4 км/ч.

let $$v$$ be the speed of the boat in still water. the speed against the current is $$v-4$$ km/h, and the speed with the current is $$v+4$$ km/h. the time taken to go against the current is $$\frac{117}{v-4}$$ hours, and the time taken to return is $$\frac{117}{v+4}$$ hours. we are given that the return trip took 2 hours less than the trip against the current. so, $$\frac{117}{v-4} - \frac{117}{v+4} = 2$$. solving this equation for $$v$$, we get $$117(v+4) - 117(v-4) = 2(v-4)(v+4)$$. this simplifies to $$117v + 468 - 117v + 468 = 2(v^2 - 16)$$. so, $$936 = 2v^2 - 32$$. $$2v^2 = 968$$. $$v^2 = 484$$. $$v = \sqrt{484} = 22$$ km/h.