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We need to optimize the distance traveled by splitting up the route into two parts: sand and water. Since it's less efficient to travel in water, it makes sense to travel less on it. C is the quickest path because it is almost a straight line while keeping the water part short.

Before I begin I want to point your attention towards Fermat's Principle of Light: The path taken between two points by a ray of light is the path that can be traversed in the least time. Consider the Brilli as pt.2 and yourself as point 1. Now the beach and the sea represent two media. If light were to travel from pt. 1 to pt. 2 and through these two media , it would surely take the least time. Now we all know that the path taken by light is similar to only one option resembling the options.

Obviously, we want to spend less time in water, so option B is obviously worse than option C, so C can't be the answer. Likewise, option A is worse than option C. Option D looks like it enters the water at the same point as option C, but it is a curve. Since the shortest distance between two points is a straight line, option D is unnecessarily long. Thus, the answer must be C

Not enough imformation to answer. We need to know how much faster on land than in water. If the difference is continually reduced then at some point option A will be the fastest.

Brachistochrone

My Labrador Retriever always gets this one right. In a field trial she is supposed to do A (straight line across land and water), but she is too smart for that and does C (the fastest way)

We can simply think it as an light particle refracting between the two surfaces.Then, answer is either B or C.Since,speed in sand is greater,I will move more distance on the sand.Hence,C is the correct answer.

Like light propagates in the medium，quentions like these all obey Snell law

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Sin i / Sin r = v1/ v2. We should do as light does.

Two words 'Snell's law ' On finding the answer to this solution you actually derive Snell's law

Consider the situation for light. You will then apply snell's law. That would be due to different refractive indices. Similarly, here instead of refractive index, we compare speed. The speed in water will be lesser than in sand. The rest is self explanatory. Correct answer being C

When the difference of speeds is maximal in favor of running, the lifeguard will run to the point closest to tbe ant, to minimize tbe time of swimming. There is no solution shown for this, though this can be seen as a limit of C.

In the opposife situation, when swimming is faster tban running as if using a boat, the lifeguard will run straight to tbe shore to minimize the time of running - case B. If the speeds are equal, the the guard will run and swim in a continuous straight line - case A.

Knowing the ratio between the speeds we could calculate the angle of C.

C is the solution.

The question isn't complete. It doesn't give how much faster it is to travel in the sand. Imagine if traveling in the sand increases your speed by an infinitesimal or extremely small amount. You would want to take the straight line. Therefore, w/o specific details on how drastically the sand increases your speed the answer could be either A or C. I'm not positive about this so if you agree or disagree please let me know

Its the same that happens when light refracts. The question is not asking the minimum distance, it is asking the minimum time. So, when light refracts, then also, it does not take the distance with the minimum length but it takes the distance that takes least time to travel.

We have two medium sand and water. We can move fast on sand than water. So we should need to cover maximum and shortcut distance from sand than after left part from water.

Since you travel fastest over sand, you want to optimize distance run on sand instead of water. C has the longest stretch of sand to run over. Thus, it is the fastest.