Now that it is summer, many of us will spend time at the beach. At times, swimmers get too far out and get into trouble. Quick action by the lifeguards is needed. They are trained to know the quickest way to get to the desperate person. Consider this challenging situation.

A lifeguard is stationed 30 meters from the shoreline. Calls for help come from a swimmer 40 m from the shoreline. The swimmer is 50 m farther down the shoreline. The guard knows they can run faster on shore than they can swim. The guard heads for the shoreline at some value of **X** they think will optimize their rescue effort. The optimal value for **X** will get them to the swimmer in the least amount of time.

During training, the lifeguard learned they could run at a speed of 4.5 m/s and swim at a speed of 2.4 m/s. Assume the swimmer stays at the same location and there is no current to complicate matters. Given these speeds and distances, what value of **X** will allow the guard to reach the swimmer in the least amount of time?

Feel free to ask clarifying questions. You do not need to commit to an answer in a comment. Do give it some thought. A solution will be published soon.

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Whew, it’s been a long time since I’ve done problems like this. On first thought I want to somehow use the Pythagorean Theorem and maximize the top angle so that as much of the distance is covered on land as possible. Am I on the right track? . . . or headed in the wrong direction?

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If you maximize the land distance, doesn’t that make X 50? Then you can use the theorem to get the running distance. After that, you would have the guard swim 40. Is that what you mean? How much time is that?

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Yes

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[…] you have an answer? A few days ago a challenge problem was offered here. It involved a lifeguard at a beach who wanted to reach a struggling swimmer in the quickest amount […]

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