A woman is standing on a cliff watching a motorboat through a telescope as the b

Question

A woman is standing on a cliff watching a motorboat through a telescope as the boat approaches the shoreline directly below her. The telescope is 250 feet above the water level and the boat is approaching at 20 feet per second. As the boat approaches the shoreline, the angle of the telescope has to change in order to keep the boat in sight. Answer both a and b below (they are two separate related rates problems). Include appropriate units. Use the figure. How fast is the distance from the telescope to the boat changing when the boat is 175 feet from the shore? If the telescope is always kept aimed at the boat, how fast is the angle of the telescope changing when the boat is 175 feet from the shore?

Explanation / Answer

Lets take the angle of the telescope with the vertical as and the distance of the boat from the shore as x.
By trigonometry, we find that
tan = x / 250
Differentiating with respect to time,
sec² (d/dt) = (dx/dt) / (250) ...(1)
At the given positions, x = 175, tan = 175/250 = 7/10 this implies, = 35°
Also, dx/dt = 20 ft/sec

Putting these values in (1)
(d/dt) = 20 / (250 sec^2 ) = 0.065 rad/sec

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