#### Answer

(a) The dolphin should head at an angle of $30.0^{\circ}$ north of west.
(b) It takes 548 seconds for the dolphin to swim home.

#### Work Step by Step

(a) In order to swim due west, the dolphin must head at an angle $\theta$ north of west such that the north component of the dolphin's velocity (relative to still water) is equal in magnitude to the south component of the water current. We can find $\theta$:
$(4.00~m/s)~sin~\theta = (2.83~m/s)~sin~45^{\circ}$
$sin~\theta = \frac{(2.83~m/s)~sin~45^{\circ}}{4.00~m/s}$
$\theta = sin^{-1}(\frac{(2.83~m/s)~sin~45^{\circ}}{4.00~m/s})$
$\theta = 30.0^{\circ}$
The dolphin should head at an angle of $30.0^{\circ}$ north of west.
(b) We can find the west component $v_x$ of the dolphin's velocity:
$v_x = (4.00~m/s)~cos~30.0^{\circ}-(2.83~m/s)~cos~45^{\circ})$
$v_x = 1.46~m/s$
We can find the time it takes to swim $0.80~km$:
$t = \frac{d}{v_x} = \frac{800~m}{1.46~m/s} = 548~seconds$
It takes 548 seconds for the dolphin to swim home.