Minimum distance between two boats travelling in different velocities
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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$\begingroup$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.
$\endgroup$ $\begingroup$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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