Suppose a cheetah is dashing due south at a rate of 30 m/s toward a popular watering hole, and a misguided gazelle is running due east toward the same spot at a rate of 20 m/s. The cheetah begins from a distance of 100 m, while the gazelle begins from a distance of 80 m. Find equations for the position y(t) of the cheetah, x(t) for the position of the gazelle, and the distance r(t) between them, and use it to find the rate of change of the distance between them at t = 0, t = 2, t = 3, and t = 4. What is the minimum distance between them? Does this occur before or after the first of them has reached the watering hole?
The method of implicit differentiation is often applied to related rates problems involving distances.