Knowledge
  • Knowledge
can someone help me on the question 2C-10. It is Unit 2 Exercises Problem Set 4. The question can be found at this link http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-2-applications-of-differentiation/part-b-optimization-related-rates-and-newtons-method/problem-set-4/MIT18_01SC_pset2prb.pdf. I have problems coming up with the basic function to represent time.
MIT 18.01 Single Variable Calculus (OCW)
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SOLVED
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jamiebookeater
  • jamiebookeater
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phi
  • phi
The solution is posted... However, they did manage to confuse the sin with the cosine The relation you will find for this problem is \[ \frac{\cos \alpha}{\cos{\beta}} = \frac{5}{2} \] In Snell's Law, they define the angle with respect to the surface normal, i.e. 90-\(\alpha\)), and you end up with sin (rather than cos) as for setting up the equations, the most basic idea is rate * time = distance and time= distance/rate Use Pythagoras to find the length of the hypotenuse of the triangle (from A to the water line). Divide this distance by the speed = 5 m/s Call this t1 Do the same for the other triangle, but use speed = 2 m/s, and call this t2 Total time is t1+t2, and it is a function of x (distance from pt P, where you dive into the water): \[ T(x) = t_1(x) + t_2(x) \] Take the derivative with respect to x, and set equal to zero, to find the critical point: \[\frac{d}{dx} T(x) = 0 \\\frac{d}{dx}t_1(x) + \frac{d}{dx}t_2(x) = 0 \]

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