2. A man in a rowing boat is presently 6km from the nearest point A on the shore. He wants to reach as soon as possible a point B that is a further 20km down from the shore from A. If he can row at 8km/hr and run at 10km/hr , how far from A should he land.

Firstly, let's look at the answer that is sought:

Let

*S* be the starting point for the rowing boat, which is 6 km from point

*A* on the shore, which is the closest point on the shore to

*S*.. We know that

*B* is 20 km from

*A* along the shore, and let point

*P* be located a distance of x km from A, so that the distance

*PB* is (20 -

*x*) km.

The attached file contains this information shown in a diagram. We seek the position of

*P* (and thus the value of

*x*) so as to minimise the travel time from

*S* to

*B*. Let

*T* be the travel time (in hours) from

*S* to

*B*, made up of the travel time from

*S* to

*P* (

*t*_{SP}) and the travel time from

*P* to

*B* (

*t*_{PB}). In otherwise, we need to minmise:

Defining the distances SP and PB (in km) as

*d*_{SP} and

*d*_{PB}, respectively, and remembering that distance equals speed times times, we can see that:

Now, applying Pythagoras' Theorem to triangle

*SAP*, we can see that

*SA*^{2} +

*AP*^{2} =

*SP*^{2}:

To find stationary points, we need to set this derivative to zero:

So, there is a stationary point at (

*x* = 8 km,

*T* = 2 h 27 min)

Following the path

*SAB* (corresponding to

*x* = 0), the rower rows 6 km taking 45 min and then runs 20 km taking 2 h, so that T = 2 h 45 min.

Following path SB (rowing directly from S to B, corresponding to

*x* = 20), the distance rowed is (6

^{2} + 20

^{2})

^{0.5} = 20.8806... km taking a little over 2 h 36 min. It follows that the stationary point is a minimum, and thus the rower should row to point

*P*, 8 km from

*A* towards

*B*.

Of course, that

*x* = 8 corresponds to a minimum can also be established by the usual methods of examining the behaviour of

*dT/dx* in the vicinity of

*x* = 8 or by showing that the second derivative is positive.

Note, however, that we have assumed that the shore line from

*A* to

*B* is a straight line with

*AB* perpendicular to

*AS*. I believe that this is what is intended, but the question does not actually specify this and if the shoreline is different from this then the answer may change.