Answer
(a) The boat should head in a direction of $28.4^{\circ}$
north of east.
(b) The velocity of the boat relative to the Earth is 3.7 m/s toward the east.
(c) It would take 135 seconds to cross the river.
Work Step by Step
(a) The river flows 2.0 m/s to the south. For the boat to move directly east across the river, the north-south component of the relative velocity of the boat to the river should be 2.0 m/s toward the north.
The relative velocity of the boat to the river is 4.2 m/s. We can find the angle $\theta$ north of east that the boat should move.
$sin(\theta) = \frac{2.0~m/s}{4.2~m/s}$
$\theta = sin^{-1}(\frac{2.0}{4.2}) = 28.4^{\circ}$
The boat should head in a direction of $28.4^{\circ}$
north of east.
(b) $v_{B/E} = (4.2~m/s)~cos(28.4^{\circ})$
$v_{B/E} = 3.7~m/s$ toward the east
The velocity of the boat relative to the Earth is 3.7 m/s toward the east.
(c) $t = \frac{width}{v} = \frac{500~m}{3.7~m/s}$
$t = 135~s$
It would take 135 seconds to cross the river.
