A high-speed motor torpedo boat is moving parallel to a straight shore line at 40 miles per hour (2/3 miles per minute), 1 and 1/2 miles from the shore, and is followed by a search-light beam which is trained on the boat from a station half a mile back from the shore. At what rate in radians per minute must the beam turn in order to follow the boat just as the boat passes directly opposite the station?
So, $$\operatorname{tan} \theta = \frac{x}{2}$$
and
$$(\operatorname{sec} \theta)^2 \frac{d \theta}{dt}= \frac{1}{2}\frac{d x}{dt}$$
and
$$\frac{d \theta}{dt} = \frac{1}{2}(\operatorname{cos} \theta)^2\frac{d x}{dt}.$$
If the boat is directly opposite the station, $\theta = 0$ degrees, and $(\operatorname{cos} 0)^2 = 1$ and
$$\frac{d \theta}{dt} = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3}.$$
Note that I converted 40 miles per hour to 2/3 miles per minute.
So, the light beam must turn 1/3 radians per minute to follow the boat as it passes directly opposite the station.
However, the book has written that the correct answer is $\frac{4}{9}.$ The book has given incorrect answers before. Is my answer incorrect, and, if so, where did I go wrong?
Thanks !
