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

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Time to row:
$t_1= \dfrac{s}{r}$

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

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