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Question: Boat $A$ was $100$ km south of boat $B$ at $9:00$ AM. If boat $A$ travels towards the north with the speed of $20$ km/h, and boat $B$ travels towards the east with the speed of $15$ km/h, when were they nearest to each other?

PS: I don't need the answer to the problem, I just need someone to guide me on how to answer this. Any leads appreciated!

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    $\begingroup$ Hint: in both answers so far, they tell you to use calculus to minimise the distance. But you will find that it is much easier to minimise the square of the distance, and then take its square root. A handy trick to remember! $\endgroup$
    – TonyK
    Oct 5, 2018 at 0:09

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If $f(t)$ is the distance between the two boats at time $t$, then you want to find the minimizer of $f$. So your task is to a) write down what $f(t)$ is using the information given in the problem, and b) find which $t$ minimizes $f$ using standard calculus techniques.

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When doing optimization problems, first identify the quantity that is to be optimized. Usually you can find it by looking for a superlative like “most” or “best” or some other “-est”. In this case, the superlative is nearest. Nearest means minimum distance. So distance between the boats is our dependent variable.

To find the independent variable, look again at the question: “when were the boats nearest to each other? That describes a moment of time.

Therefore, express the distance between the boats as a function of time. Use calculus to minimize that function.

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