Question

A child in danger of drowning in a river is being carried downstream by a current that flows due south uniformly with a speed of 3 km/hr. the child is 0.5 km from the shore and 2 km north (i.e., upstream) of a boat landing when a rescue boat sets out.a) if the boat proceeds at its maximum speed of 24 km/hr with respect to the water, what angle relative to north should the pilot point the bow of the boat in order to reach the child in the minimum amount of time? please enter your answer in degrees.angle james walker

Answer 1

Let the rescue boat starts at an angle theta with the North

now its velocity towards East is given as

$v_x = 24sin\theta$

$v_y = -24cos\theta + 3$

now in some time "t" it will catch the boy

so we will have

$t = \frac{0.5}{24sin\theta}$

also we have

$t = \frac{2}{-24cos\theta + 3}$

now we have

$\frac{2}{-24cos\theta + 3} = \frac{0.5}{24sin\theta}$

$4*24sin\theta = - 24cos\theta + 3$

$96 sin\theta + 24cos\theta = 3$

by solving above we got

$\theta = 164 degree$