# 19 Direction of the man to reach his destination as soon as possible

**Problem 19**

A man on an island *a* miles south of a straight beach wishes to reach a point on shore *b* miles east of his present position. If he can row *r* miles per hour and walk *w* miles per hour, in what direction should he row, to reach his destination as soon as possible? See Fig. 57.

**Solution 19**

$t_1= \dfrac{s}{r}$

Time to walk:

$t_2 = \dfrac{b - x}{w}$

Total time:

$t = t_1 + t_2$

$t = \dfrac{s}{r} + \dfrac{b - x}{w}$

From the figure:

$x = a \tan \theta$

$s = a \sec \theta$

$t = \dfrac{a \sec \theta}{r} + \dfrac{b - a \tan \theta}{w}$

$\dfrac{dt}{d\theta} = \dfrac{a \sec \theta \, \tan \theta}{r} - \dfrac{a \sec^2 \theta}{w} = 0$

$\dfrac{a \sec \theta \, \tan \theta}{r} - \dfrac{a \sec^2 \theta}{w} = 0$

$\dfrac{\tan \theta}{r} - \dfrac{\sec \theta}{w} = 0$

$\dfrac{\sin \theta}{r \, \cos \theta} - \dfrac{1}{w \, \cos \theta} = 0$

$\dfrac{\sin \theta}{r \, \cos \theta} = \dfrac{1}{w \, \cos \theta}$

$\sin \theta = \dfrac{r}{w}$ *answer*