Related rates running distance problem

During a night run, an observer is standing 80 feet away from a long, straight fence when she notices a runner running along it, getting closer to her. She points her flashlight at him and keeps it on him as he runs. When the distance between her and the runner is 100 feet, he is running at 9 feet per second. At this moment, at what rate is she turning the flashlight to keep him illuminated? Include units in your answer.

Related rates problems involve determining how one quantity changes in relation to another over time. To approach these problems, begin by carefully identifying the variables and the relationships between them. The goal is usually to find the rate of change of one variable by using an equation that connects it to other variables whose rates of change are known. Once you have an equation that represents the situation, take its derivative with respect to time, applying the chain rule as needed. Finally, substitute the known values to solve for the unknown rate.

In this problem, the observer is tracking the runner with her flashlight, which creates a geometric relationship between the observer, the runner, and the flashlight angle. The situation can be modeled using right triangle relationships, with the distance between the observer and the runner being the hypotenuse, and the distance between the observer and the fence being one leg of the triangle. By differentiating the equation that relates the angle of the flashlight to the positions of the observer and the runner, you can find the rate at which the observer is turning the flashlight in radians per second, based on the runner's speed and the current distances.

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