MATH100 December 2018
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Particle travels with a constant speed of 2 units per minute on the -axis starting at the point (4,0) and moving away from the origin, while particle travels with a constant speed of 1 unit per minute on the -axis starting at the point (0,8) and moving towards the origin. Find the rate of change of the distance between the two particles when the distance between the two particles is exactly 10 units.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Use Pythagoras' theorem to relate the positions of the two particles.
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Let be the distance from the origin to the particle and let be the distance from the origin to the particle . Also Let be the distance between and . Notice that and are functions of time and we shall thus write them as respectively.
By the given information, we have units/min and units/min. Using the initial positions, then we see that and . We can relate these two functions by Pythagoras' theorem: we have , and therefore
Now, we will first determine the time at which the distance between the particles is units. We do this by setting and solving for . This gives
and therefore (we can ignore the negative root since time cannot be negative).
Next, we want to find when . By differentiating both sides of the equation , we get . Now we can plug in and , which gives .
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