Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. consider a long stretch of river flowing north, with parallel banks 40 m apart. if the maximum water speed is 3 m/s, a reasonable model for the water speed of a river x units from the west bank is a sine function: f(x) = 3 sin (πx/40).if a boater would like to cross theriver from a point on the west bank and land at a point exactly opposite with constant headin and a constant speed of 5 m/s, determine the angle at which the boat should head.

The angle at which the boat must head is

Solution:

As per the solution:

Distance between the parallel banks, d = 40 m

The maximum speed of water, v' = 3 m/s

constant speed, u' = 5 m/s

Also,

The speed of water of the river at a distance of 'x' units from the  west bank is given as a sine function:

         (2)

Now, to determine the angel at which the boat must head:

The velocity of the engine of the boat:

v =

v =

The abscissa of the boat at time t:

v =

Now, from above and eqn(1) , we can write:

Now, boat's velocity at time t:

v =

In order to obtain the position of the boat, we integrate both the sides, we get:

r = + C             (3)

Now, at r = 0:

0 = + C

C =

Now, from eqn (3)

r =                      (4)

the baot will reach the point at y = 0 and x = 40

Now,

40 =

Substituting the above value of 't' in eqn (4):

r =

We get:

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