You need to add up the time taken for him to run on land and the time taken for him to swim up the river.
If he runs at 2 km/h, then the time taken will be give by
, so
. So we need the total distance that he runs, and then divide it by his speed. Since he is running from his camp at
)
to a point on the river
)
, which since we know it is on that curve is
)
, we can use the distance formula,
^{2}+(y_{2}-y_{1})^{2}})
, or you can just look at it using pythagoras. Anyways that gives:
^{2}+\left(x^{2}-1\right)^{2}}<br />\\ & =\sqrt{x^{4}-2x^{2}+1+x^{2}}<br />\\ & =\sqrt{x^{4}-x^{2}+1}<br />\\ t_{1} & =\frac{d_{1}}{s_{1}}<br />\\ & =\frac{\sqrt{x^{4}-x^{2}+1}}{2}<br />\end{alignedat})
Now for the time it take him to swim. We are told that this time is 'proportional to the difference between the
coordinates of the desalination plant (which is
) and the point where he enters the river (which is
).
So since it is 'proportional to', we have to introduce the constant of proportionality
. If something is proportional to something else, then as the first changes the second will change by a scalar amount according to how it is proportional to the first. i.e. If
is directly proportional to
, that is
, then
, where
. If say for example we had
is proportional to the square of the
coordinate,
we would have
, so
changes a certain amount for each change in
.
The difference between the
coordinates is
 & =\frac{3-4x^{2}+4}{4}<br />\\ & =\frac{1}{4}\left(7-4x^{2}\right)<br />\end{alignedat})
)
As
is proportional to this we have,
)
.
Now the sum of those two gives us the total time taken, hence
)
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