I give my calculus students a problem every time I teach them, and it takes a lot of time to do it with half the techniques. At the end of the exercise, I ask them if they would like to see a three to five line solution that a smart fifth grader would understand. They all want to see it, so I give them a scenario: It is a hot October day in California about 200 years ago and there is a poor cow with a broken leg 6 kilometers away from a river. A farmer is 4 kilometers away from the cow and 2 kilometers away from the river, and he has a bucket. I ask them what the farmer needs to do to get the cow water in the shortest amount of time. It looks simple, but the calculus solution is not easy to solve. It is hard to figure out when the derivative is non-negative without doing any calculus.But, if you know calculus, it is easy to see that the derivative is non-negative when $x \geq 1$. Therefore, the farmer should go up to 1 kilometer, dip the bucket, and go to the cow. So we have a similar triangle, we have two sides that are proportional, and we have the third side which is the same.
We can simplify the problem by creating a problem that looks almost identical to the original problem, but is actually trivial to solve. What makes the problem hard is that the farmer and the cow are on the same side of the river. We can make our problem look like this simple one by using a reflection operation to flip the farmer across the river. The phantom farmer must go directly to the cow, and this point will be our desired point x. We claim that the real farmer must go to the same point x, and we can prove this by noting that any path taken by the real farmer is mimicked by the phantom farmer. We can also prove that the minimal path for the phantom farmer is the minimal path for the real farmer by using the triangle inequality. Finally, we can find the point x without calculus by noting that triangle FAX is congruent to triangle F’AX, and those are similar to Triangle CBX. The calculus is more flexible, you can use it in more situations.But the geometry is more visual and it’s more direct, so it’s easier to understand. For this particular type of problem, the geometrical solution is definitely more elegant. However, if the problem is made more complicated with additional variables such as a non-straight river, or the farmer having to get food for the cow from somewhere else, then the calculus solution may be necessary. Furthermore, there is a folklore problem with both a calculus solution and a super sweet geometry solution. Links to this problem and other videos with Zvezda, plus a podcast where she tells her life story, can be found in the video description.