Answer
a) $50\;\rm m$
b) See the figure below.
Work Step by Step
a) First of all, we need to draw the two velocity vectors of the river and the boat, as you see in the first figure below.
As we see, when the boat is directed perpendicular to the river (toward the North), the water speed will make the boat drift toward the east.
The boat's northward velocity component is $v_b$ and its eastward velocity component is $v_r$ which is the speed of the current in the river.
The width of the river is 100 m, so the boat will take this distance in 50 s.
$$v_b=\dfrac{\Delta y}{\Delta t}$$
$$ \Delta t =\dfrac{\Delta y}{ v_b}=\dfrac{100}{2}=\bf 50\;\rm s$$
Now we know that the board will be drifted to the east due to the water current at a constant speed of 1 m/s.
So, the eastward distance traveled by boat is given by
$$\Delta x=v_r\Delta t=1\cdot50=\color{red}{\bf 50}\;\rm m$$
b) See the figure below.
