At time t = 0, an insect starts crawling along a straight line at a rate of 3 ft/min. Two minutes later, a second insect starts from the same position, crawling in a direction perpendicular to that of the first and at a speed of 5 ft/min. How fast is the distance between them changing once the first insect has traveled 12 feet?

Difficulty rating: 

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Solution:

        Once the first insect has crawled 12 feet (at a rate of 3 ft/min), 4 minutes will have elapsed. Remember that the second insect begins traveling two minutes after the first, so it only gets to travel 2 minutes, and therefore 10 feet (since his rate is 5 ft/min). We now have a right triangle with lengths 12 and 10 and an unknown hypotenuse length. I will call the first insect x and the second y; meanwhile, the hypotenuse I will call d. (I will shortly find length d using the Pythagorean theorem.) Because I have a right triangle, the Pythagorean theorem will be my starting point, as it gives a relationship between all the variables. Remember that we need to take the derivative of everything with respect to time when doing related rates, so that is my next step.

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